Anantha Krishna Karthik-Thesis

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BLIND MODULATION CLASSIFICATION AND SYMBOL RATE

ESTIMATION IN RAYLEIGH FADING MULTIPATH ENVIRONMENT

WITH FREQUENCY AND TIMING OFFSETS

A thesis submittted in partial fulfillment of

the requirements for the degree of

Master of Science (by research)

In

COMMUNICATION SYSTEMS AND SIGNAL PROCESSING

by

Anantha Krishna Karthik N

200531002

[email protected]

Communication Research Center

INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY,

GACHIBOWLI, HYDERABAD, A.P. INDIA - 500 032

May 2011

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INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY

GACHIBOWLI, HYDERABAD, A.P., INDIA - 500 032

CERTIFICATE

It is certified that the work contained in this thesis, titled Blind Modulation Classification

and Symbol Rate Estimation in Rayleigh Fading Multipath Environment with Frequency

and Timing Offsets by Anantha Krishna Karthik N has been carried out under my su-

pervision and it is fully adequate in scope and quality as a dissertation for the degree ofMaster of Science

Date Dr. V. U. Reddy (Advisor)

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Abstract

Blind modulation classification is the process of identification of the modulation format

of the transmitted signal using only the samples of the received signal. In this thesis, we

address two topics related to blind modulation classification. The first topic deals with

blind symbol rate estimation. A reliable estimate of the symbol rate irrespective of the

modulation format of the transmitted signal is essential for blind modulation classification.

We present a method based on the cyclic autocorrelation function for blind estimation of

the symbol rate of a linear digitally modulated signal propagated through multipath in the

presence of timing and carrier frequency offsets. The performance of the proposed symbol

rate estimation algorithm is evaluated under the above conditions through simulations.

The second topic of the thesis deals with the problem of modulation classification.

We propose and evaluate a feature-based hierarchical modulation classification method,

which is developed to discriminate between the various modulation formats in the presence

of Rayleigh fading multipath and, timing and carrier phase offsets. In our feature-based

modulation classification method, we first estimate the overall impulse response (inclusive of

both the pulse shape and multipath channel) using only second-order statistics and then use

the estimated channel to compute the various features of interest. We use both cumulants

and moments as features to discriminate between the various modulation formats. We use

a hierarchical structure in which lower-order features are used in the earlier stages and

higher-order features in the later stages. Performance of the proposed classification method

is evaluated under Rayleigh fading flat as well as multipath environment in the presence of

timing and carrier phase offsets using simulations.

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Acknowledgement

There are many people i would like to thank for their support and knowledge without whom

this research would not have been possible. Firstly, to my thesis advisor Dr. V. Umapathi

Reddy, thank you for your enthusiasm, patience and guidance. I would also like to express

my sincere appreciation and gratitude to my family and friends for their constant support

and encouragement.

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Contents

Abstract iii

Acknowledgement iv

1 Introduction 1

1.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Symbol rate estimation . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Modulation classification . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Symbol Rate Estimation 13

2.1 Basics of Cyclostationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Cyclostationarity in Linear Digitally Modulated Signals . . . . . . . . . . . 15

2.2.1 Sampling of the continous-time signal . . . . . . . . . . . . . . . . . 18

2.3 Cyclostationarity in Signals Propagated Through Multipath . . . . . . . . . . . . 18

2.3.1 Cyclic autocorrelation function of the sampled received signal . . . . 19

2.4 Symbol Rate Estimation Algorithm and Simulation Results . . . . . . . . . 21

2.4.1 Algorithm for symbol rate estimation . . . . . . . . . . . . . . . . . 21

2.4.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.3 Selection of the lag parameter . . . . . . . . . . . . . . . . . . . . . . 232.4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Classification in Multipath Environment 28

3.1 Subspace Based Method for Channel Identification . . . . . . . . . . . . . . 29

3.1.1 Non-identificable channels . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Features Used in the Classification . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Definitions of the features used . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Relation between the features ofsn and those off(n) . . . . . . . . 36

3.3 Algorithm for Hierarchical Modulation Classification . . . . . . . . . . . . . 37

3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 Flat fading Rayleigh channel . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Conclusion 46

4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 48

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List of Tables

2.1 Multipath physical channel profile . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Percentage of successful estimation of symbol rate for non-uniformly spaced multi-

path channel at an SNR of 5 dB (Average corresponds to the success rate, averaged

over all the constellations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Percentage of successful estimation of symbol rate for uniformly spaced multipath

channel at an SNR of 5 dB (Average corresponds to the success rate, averaged over

all the constellations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Percentage of successful estimation of symbol rate for non-uniformly spaced multi-

path channel at an SNR of 0 dB (Average corresponds to the success rate, averaged

over all the constellations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Percentage of successful estimation of symbol rate for uniformly spaced multipath

channel at an SNR of 0 dB (Average corresponds to the success rate, averaged over

all the constellations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Values of the features of interest for the underlying constellations . . . . . . . . . 36

3.2 Percentage of correct classification in Rayleigh fading multipath physical channel

(tap variances is as given in Table 2.1, N= 2000 and 10,000 trials) . . . . . . . . 41


(tap variances is as given in Table 2.1, N= 4000 and 10,000 trials) . . . . . . . . 42

3.4 Multipath physical channel profile . . . . . . . . . . . . . . . . . . . . . . . . . 42


(tap variances as given in Table 3.4, N= 4000 and 10,000 trials) . . . . . . . . . 44

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3.6 Percentage of correct classification in Rayleigh fading flat physical channel (fading

coefficient of unit variance, N= 4000 and 10,000 trials) . . . . . . . . . . . . . . 45

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List of Figures

2.1 Percentage of successful symbol rate estimation for various lag values (= 0.5, pulse

shape = root raised cosine). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Percentage of successful symbol rate estimation for various lag values ( = 0.35,

pulse shape = root raised cosine). . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Algorithm for hierarchical modulation classification (modulations considered are

PAM, PSK and QAM constellations) . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Quality of the estimated channel in Rayleigh fading multipath physical channel case 43

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Chapter 1

Introduction

Blind modulation classification is the process of identification of the modulation format of

the transmitted signal using only the samples of the received signal. It is the intermediate

step between signal detection and demodulation, and has applications in both cooperative

and non-cooperative environments such as software-defined radio, cognitive radio, surveil-

lance and electronic warfare. In the presence of various practical problems such as carrier

frequency offset, timing offset and multipath fading environment, blind modulation classifi-

cation is a challenging task. In blind modulation classification, another important parameter

that needs to be estimated is the symbol rate. A reliable estimate of the symbol rate, in the

presence of various practical problems indicated above, is essential for proper modulation

classification.

1.1 Brief Review of the Related Work Reported in the Lit-

erature

1.1.1 Symbol rate estimation

Research in the area of blind modulation classification as well as blind symbol rate estima-

tion has been going on for many years. A number of algorithms have been developed for the

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CHAPTER 1. INTRODUCTION 2

blind estimation of the symbol rate. Algorithms based on the cyclic autocorrelation func-

tion [1]-[4], wavelet transforms [5]-[6] and the Fourier transform [7] are the most frequently

used approaches for symbol rate estimation.

The cyclic autocorrelation approach for symbol rate estimation was proposed in [1]. The

basic idea behind the cyclic autocorrelation approach is that the autocorrelation function of

a linear digitally modulated sequence is a periodic function in time with a period equal to the

symbol period. The author uses this fact to express the autocorrelation function as a Fourier

series and extract the symbol period from the coefficients of the Fourier series expansion.

The proposed method however has poor performance for raised cosine pulse shapes having

small roll-offs ( 0.3), when no coarse estimate of the symbol rate is available a priori.

Dandawate and Giannakis [2] modified the method of [1] and proposed the use of a weighing

matrix in computation of the cyclic autocorrelation function to detect the cycle frequencies

for pulse shapes having small roll-offs in the absence of carrier frequency offset.

Mazet and Loubaton [3] used the concept proposed in [2] to estimate the symbol rate of

a linear digitally modulated signal. They assumed perfect carrier frequency synchronization

and a root raised cosine pulse shape. They declared their symbol rate estimate a success if

the estimation error was less than 1% of the actual symbol rate. They obtained a success

rate of 99.2% for the QPSK symbol constellation with a roll-off factor 0.2 for data length

of 1000 symbols at an SNR of 60 dB with oversampling factor of 4. The method proposed

in [3] suffers from high computational complexity and the performance degrades at lower

SNRs.

In [4], the authors proposed a modified cyclic autocorrelation approach for the symbol

rate estimation of M-PSK signals. They have shown that for M-PSK signals using the

rectangular pulse shape, the cyclic autocorrelation value reaches the maximum when the

lag value is chosen as half of the symbol period. They assumed a line-of-sight (LOS) channel,

rectangular pulse shape and knowledge of coarse estimate of the symbol rate. They quoted

a root mean square error of 0.06 and 0.14 for PSK-2 and PSK-4 symbol constellations,

respectively, for data length of 600 symbols at an SNR of 0 dB with oversampling factor of

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Chan et al. [5] proposed the use of wavelet transform for estimating the symbol rate of

M-PSK signals. They used the discontinuities in the magnitude of the wavelet transform

of the received signal to extract the symbol rate. They chose the discrete Haar wavelet

to compute the wavelet transform. In their work, they assumed perfect carrier frequency

synchronization, rectangular pulse shape and a LOS channel. They obtained a mean square

error of 104 for the {PSK-2, PSK-4, PSK-8}symbol constellations for data length of 100

symbols at an SNR of 7 dB with oversampling factor of 3.

Yu et al. [6] proposed a method based on filter banks for symbol rate estimation. In

this method, the authors extracted a coarse estimate of the symbol rate from the spectrum

of the signal and then estimated the actual symbol rate using a combination of a filter

bank and a fourth-order nonlinearity unit. They assumed a LOS channel, perfect carrier

frequency synchronization and a root raised cosine pulse shape. They declared their symbol

rate estimate a success if the error in the estimate was less than the corresponding DFT

resolution. They quoted a success rate of nearly 100% for the BPSK symbol constellation

with roll-off factor 0.2 for data length of 4096 symbols at an SNR of 0 dB with oversam-

pling factor of 4. Unlike the approach of [1]-[3], the proposed algorithm is applicable to

pulse shapes having zero roll-off. However, the algorithm suffers from high computational

complexity and might fail in multipath environments.

The FFT-based approach for symbol rate estimation was proposed in [7]. A coarse

estimate of the symbol rate is extracted from the power spectrum of the received signal,

and based on the coarse estimate a least squares formulation is used to improve the accuracy

of the estimate. The authors assume LOS channel and a raised cosine pulse shape with roll-

off factor 0.4.

1.1.2 Modulation classification

The approaches followed in blind modulation classification can be broadly divided into

two groups: decision-theoretic approach and feature-based approach. Decision-theoretic

approaches [13]-[17] treat the modulation classification problem as a multiple hypothesis

testing problem. The decision-theoretic classifiers with the maximum likelihood tests are


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optimal, but the corresponding closed-form solutions are either unavailable or involve nu-

merical search of high computational complexity. This approach is not robust to the model

mismatch in the presence of phase or frequency offsets, residual channel effects and so on.

On the other hand, feature-based methods rely on the features derived from the data for

modulation classification. A library of features used for classification is usually derived

off-line, and decision is made based on the best match of the features estimated from real-

time finite data with those in the library. The commonly adopted features are based on

higher-order statistics (HOS) including cumulants [18]-[22], [32]-[33], moments and cyclic

cumulants [27]-[31]. Cumulants are generally preferred due to their favorable properties

over moments [8]. In contrast to the decision-theoretic methods, the feature-based methods

are non-optimal, but they are simple to implement and can often yield performance close to

the optimal, if carefully designed. A survey of the work done in the area of blind modulation

classification was reported in [34].

Huang and Polydoros [13] proposed a maximum-likelihood method for classifying M-

PSK signals in an AWGN (additive white Gaussian noise) channel which is similar to a LOS

channel except that the channel gain is taken as unity in the AWGN case. They developed

maximum-likelihood classifiers for both the coherent as well as the non-coherent scenarios.

In the coherent case, carrier phase/frequency and symbol timing are assumed known while

in the non-coherent case, the carrier phase is assumed unknown. They considered one

sample per symbol interval, rectangular pulse shape and data length of 100 symbols. They

provided results for various 2-class problems. For the {BPSK, QPSK} 2-class problem,

they obtained an average success rate of 97% and 95% for the coherent and non-coherent

scenarios, respectively, at an SNR of -2 dB. For the {QPSK, PSK-8} 2-class problem, the

corresponding results were 90% and 85% at an SNR of 3 dB, while for the {PSK-8, PSK-16}

2-class problem, the results were 95% and 89% at an SNR of 10 dB.

Chugg et al. [14] proposed a two-stage maximum-likelihood classifier for classifying

{BPSK, QPSK, OQPSK} constellations in an AWGN channel in the presence of random

phase. They assumed that the BPSK/QPSK constellations have a symbol rate twice that


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of the OQPSK constellation. They assumed perfect carrier frequency synchronization, rect-

angular pulse shape and one sample per symbol interval for the BPSK/QPSK symbol con-

stellations, two samples per symbol interval for the OQPSK symbol constellation. They

obtained an average success rate of 99% for data length of 1000 samples at an SNR of 4 dB.

Sills [15] developed a maximum-likelihood classifier for a 6-class problem for both the co-

herent as well as the non-coherent scenario. For both cases, he assumed an AWGN channel,

rectangular pulse shape and one sample per symbol interval. He considered the following

constellations: {BPSK, QPSK, PSK-8, QAM-16, QAM-32, QAM-64}. He obtained an av-

erage success rate of nearly 100% at an SNR of 10 dB for data length of 256 symbols in the

coherent case, while in the non-coherent case he obtained an average success rate of 87%.

Wei and Mendel [16] proposed a likelihood-based modulation classifier for the following

constellations: {QAM-16, V.29, QAM-32, QAM-64}. They assumed a coherent scenario

as well as the knowledge of the noise power. They assumed a rectangular pulse shape,

one sample per symbol interval and an AWGN channel. They provided results for the

{V.29, QAM-16} 2-class problem and {QAM-16, QAM-32, QAM-64} 3-class problem for

data lengths of 100, 200 and 1000 symbols. For the 2-class problem, they obtained an

average percentage of correct classification as 100% at an SNR of 5 dB for data length of

1000 symbols. For the 3-class problem, they reported the percentage of correct classification

of nearly 90% for the QAM-16 constellation at an SNR of 10 dB for data length of 1000

symbols.

Panagiotou et al. [17] reported a likelihood-based modulation classifier for an AWGN

channel in the presence of random phase. They assumeda priori knowledge of frequency

and timing offset (equivalent to having symbol synchronization) and considered one sample

per symbol interval. They considered three 2-class problems: {QAM-16, PSK-16}, {V.29,

PSK-16}, {QAM-16, V.29}. For all the 2-class problems considered, they obtained an

average success rate of nearly 99% at an SNR of 6 dB for data length of 100 symbols.

Feature-based modulation classification has been reported in [18]-[33]. Swami and Sadler

[18] developed a modulation classification algorithm using fourth-order cumulants as the

features of interest in an hierarchical structure. They considered one sample per symbol


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interval and perfect channel equalization. They considered several 2-class problems, a 4-

class problem{BPSK, PAM-4, QAM(4,4), PSK-8}and a 8-class problem {BPSK, PAM-4,

PSK-4, PSK-8, V32, V29, V29c, QAM(4,4)}. They also briefly discussed the effects of

various channel imperfections on the classifier performance. For the 4-class problem they

provided results for three cases. The cases were i) signal corrupted by white Gaussian noise,

ii) signal corrupted by phase errors and white noise, iii) signal corrupted by frequency offset

and white noise. For these cases, they considered SNRs of 5 dB, 10 dB for data lengths of

100, 250, 500 symbols. They obtained an average success rate greater than 90% for data

lengths 250 symbols in all cases. For the 8-class problem they considered two cases: i)

SNR of 20 dB and data length of 500 symbols, ii) SNR of 10 dB and data length of 1000

symbols. For both the cases, they obtained an average success rate greater than 96%.

Liu and Xu [19] developed a feature-based modulation classification algorithm using

fourth- and eighth order cumulants as the features of interest for M-QAM, M-PSK, M-ASK

classification in a LOS channel using an hierarchical structure. They considered one sample

per symbol interval, rectangular pulse shape and perfect carrier frequency synchronization.

They considered several 2-class problems, two 3-class problems and one 5-class problem. For

the 3-class problem consisting of{ASK-4, ASK-8, ASK-16} constellations, they obtained

an average success rate of about 88% for data length of 1000 symbols at an SNR of 10 dB.

For the 3-class problem consisting of {BPSK, QPSK, PSK-8} constellations, an average

success rate of nearly 100% was obtained for data length of 1000 symbols at an SNR of

5 dB. For the 5-class problem consisting of{BPSK, ASK-4, QPSK, QAM(4,4), QAM32}

constellations, the average success rate was 94% for data length of 1000 symbols at an SNR

of 5 dB.

In [18], Swami and Sadler briefly discussed differential processing in order to combat

the problem of residual frequency offset in blind modulation classification. This idea was

incorporated in [21] where the authors develop a classification algorithm using the cumu-

lants of the differentially processed signal as the features of interest in order to perform

modulation classification in the presence of frequency offsets. They considered a 10-class


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problem consisting of the following constellations: {BPSK, PSK-8, PSK-16, PAM-4, QAM-

8, QAM-16, QAM-32, QAM-64, QPSK, OQPSK}. They considered 2 samples per symbol

interval, rectangular pulse shape and a LOS channel. They obtained an average success

rate of 90% at an SNR of 10 dB for data length of 3000 symbols.

In [22], the authors incorporated the idea of [18] and used a combination of higher-

order moments of the received signal as well as the moments of the differentially processed

signal as the features of interest to perform modulation classification in the presence of

frequency offsets. They considered a 10-class problem consisting of the following constel-

lations: {PAM-2, PAM-4, PAM-8, PSK-4, PSK-8, PSK-16, QAM-16, QAM-32, QAM-64,

OQPSK}. Choosing 2 samples p er symbol interval, rectangular pulse shape and assuming

a LOS channel, they obtained an average success rate of 91% at an SNR of 10 dB for data

length of 1000 symbols.

Much of the reported work in the area of blind modulation classification consider only

digital modulations. However, few works such as [23]-[24] address the problem of joint

analog and digital modulation classification. Nandi and Azzouz [23] developed a modu-

lation classification algorithm based on the instantaneous features of the received signal.

All the features of interest in the proposed classification algorithm were derived from the

instantaneous amplitude, phase and frequency of the received signal. They assume a priori

knowledge of the carrier phase/frequency and a LOS channel, and a rectangular pulse shape

for the linear digital modulations. The authors divided the intercepted signal into M non-

overlapping successive frames each containing 2048 samples and applied the classification

algorithm to each frame and chose the modulation format which was declared in majority

of the M frames. They developed a hierarchical as well as a neural network structure for

a 13-class problem consisting of the following modulation formats: {AM, DSB, VSB, LSB,

USB, FM, AM-FM, PSK-2, PSK-4, ASK-2, ASK-4, FSK-2, FSK-4}. The proposed algo-

rithm gives an average success rate of nearly 94% at an SNR of 15 dB with a total of 400

frames.

Dobre et al. [24] developed a modulation classifier for a 9-class problem consisting

of both analog and digital modulations. The authors considered the following modulation


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formats: {AM, DSB, USB, LSB, BPSK, QPSK, PSK-8, QAM-16, QAM-64}. They assumea

prioriknowledge of the carrier phase/frequency and a LOS channel, and use a raised cosine

pulse shape with roll-off factor 0.25 for the linear digital modulations. In the proposed

classifier, they consider the digital modulations {QPSK, PSK-8, QAM-16, QAM-64} as

a single modulation group. The authors use a combination of spectral features and the

cyclic frequency as features of interest for modulation classification. The bandwidth of the

baseband signal is fixed at 3 KHz and the signal is sampled at 48 KHz. They obtained an

average success rate of nearly 100% at an SNR of 2 dB for data length of 48,000 samples.

Ho et al. [25]-[26] proposed the use of the wavelet transform to perform modulation

classification. They use the discontinuities in the magnitude of the wavelet transform of the

received signal to discriminate between the various modulation formats. They used the Haar

wavelet to compute the wavelet transform. They considered a 6-class problem consisting of

the following modulation formats: {PSK-2, PSK-4, PSK-8, FSK-2, FSK-4, FSK-8}. They

used 125 samples per symbol interval and rectangular pulse shape for the PSK modulations,

assuming p erfect carrier frequency synchronization and a LOS channel. They considered

three cases: i) Inter-class classification between PSK and FSK constellations, ii) Intra-class

classification between the PSK constellations, iii) Intra-class classification between the FSK

constellations. In the first case, the scale in the wavelet decomposition was fixed as 14 and

they obtained an average success rate of 98.2% at an SNR of 13 dB for data length of 50

symbols. For the second case, they fixed the scale as 125 and obtained an average success

rate of 92.3% at an SNR of 13 dB for data length of 100 symbols. In the third case, the

scale was chosen such that the separation of the levels at different scales is maximized, and

they obtained an average success rate of 98% at an SNR of 15 dB for data length of 100

symbols.

Marchand et al. [27] proposed the use of fourth-order cyclic cumulants in order to

differentiate between the PSK-4 and QAM-16 signal constellations. They considered 10

samples per symbol interval, assumed rectangular pulse shape and a LOS channel. The

proposed algorithm gives an average success rate of nearly 82% at an SNR of 0 dB for data

length of 500 symbols.


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Spooner [28] proposed the use of sixth-order cyclic cumulants for modulation classifica-

tion. He considered three 2-class problems, two 3-class problems and two 4-class problems.

He assumed a root-raised cosine pulse shape with a roll-off factor 0.35, 10 samples per sym-

bol interval, a LOS channel and a prioriknowledge of the pulse shape coefficients. For the

{QAM-16, QAM-64}, {V.29, QAM-16} and {V.29, QPSK} 2-class problems, he obtained an

average success rate of 80%, 95% and 99%, respectively, at an SNR of 9 dB for data length of

3000 symbols. For the {QAM-16, QAM-64, V.29}and {QAM-16, QAM-64, QPSK} 3-class

problems, he obtained an average success rate of 85% and 90%, respectively, at an SNR of

9 dB for data length of 3000 symbols. For the {BPSK, QPSK, /4-QPSK, PSK-8} and

{QPSK, QAM-8, QAM-16, V.29} 4-class problems, he obtained an average success rate of

99% and 65%, respectively, at an SNR of 9 dB for data length of 6000 symbols.

Dobre et al. [29] evaluated the p erformance of fourth-, sixth- and eighth-order cyclic

cumulants for modulation classification for several 2-class problems. They considered a LOS

channel, raised cosine pulse shape with roll-off factor 0.25 and assumed a prioriknowledge of

carrier frequency as well as pulse shape coefficients. The oversampling factor was chosen so

as to eliminate cycle aliasing. They considered three 2-class problems: {QAM-16, QPSK},

{ASK-4, ASK-8} and {QAM-16, QAM-64}. For the{QAM-16, QPSK} 2-class problem,


symbols. For the {ASK-4, ASK-8} and {QAM-16, QAM-64} 2-class problems, using the

eighth-order cyclic cumulants, they obtained an average success rate of 90% and 85%,

respectively, at an SNR of 5 dB for data length of 20000 symbols .

Dobre et al. [30] developed an algorithm based on higher-order cyclic cumulants for

the modulation classification of QAM signals in the presence of carrier phase and carrier

frequency offsets. They proposed the use of a feature vector consisting of fourth-, sixth-

and eighth-order cyclic cumulants. They considered 9 samples per symbol interval and,

assumed a LOS channel and a raised cosine pulse shape with roll-off factor 0.25. They also

assumeda prioriknowledge of the pulse shape coefficients. For a 2-class problem consisting

of{QAM-4, QAM-16} constellations, an average success rate of 96% was obtained at an

SNR of 7 dB for data length of 900 symbols.


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Dobre et al. [31] proposed the use of eighth-order cyclic cumulants as the features of

interest to perform modulation classification in flat fading environments. They considered a

9-class problem consisting of the following constellations: {PAM-2, PAM-4, PAM-8, PSK-8,

PSK-16, QAM-4, QAM-16, QAM-32, QAM-64}. They assumed perfect carrier frequency

synchronization and a prioriknowledge of the pulse shape coefficients as well as the flat

fading channel coefficient. They considered 11 samples per symbol interval and a raised

cosine pulse shape with roll-off factor 0.35. The proposed algorithm gives an average success

rate of 81% at an SNR of 10 dB for data length of 4000 symbols.

The problem of modulation classification in multipath environments is a challenging

and complex problem. In [18], Swami and Sadler brought out, in asymptotic case, how

cumulants are affected in multipath environments. However, they did not suggest any

method to compensate or correct them. Wu et al. [32] suggested the use of moment based

method for blind channel estimation and then use the estimate so obtained to calculate the

various cumulant features. They assumed one sample per symbol interval, a rectangular

pulse shape and p erfect carrier frequency synchronization. They considered two multipath

channel models: i) {1, c1, c2, c3} where c1, c2, c3 are zero mean complex Gaussian random

variables, each with variance 0.05, and the spacing between consecutive channel taps is

equal to the symbol period. ii) {1, c1, c2, , c9}, wherec1, c2, , c9are zero mean complex

Gaussian random variables each with variance 0.05 and the spacing between consecutive

channel taps is equal to the symbol period. They gave simulation results for two cases, a)

{BPSK, QPSK} and b) {QAM-4, QAM-16, QAM-64}, for both the channel models. For

the 2-class problem with four-tap channel model, they obtained an average success rate of

89% at an SNR of 5 dB for data length of 250 symbols, and with the ten-tap channel model

for the same 2-class problem, they obtained an average success rate of 84% at an SNR of

5 dB for data length of 500 symbols. For the 3-class problem with four-tap channel model,


symbols, and with the ten-tap channel model for the same 3-class problem, they obtained

an average success rate of 69% at an SNR of 12 dB for data length of 4000 symbols.

In [32], the authors used only higher-order statistics for channel estimation. However,


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estimates obtained from higher-order statistics generally have larger variances compared to

those based on lower-order statistics [8]. In [33], the author uses this reason to propose a

method for channel estimation based on a combination of both second-order and higher-

order statistics. After estimating the channel, he uses the estimated channel to compute the

various features of interest in order to perform modulation classification. He considered one

sample per symbol interval, rectangular pulse shape, perfect carrier frequency synchroniza-

tion and the above mentioned four-tap multipath channel model {1, c1, c2, c3}. He provided

results for a 9-class problem consisting of the following constellations: {PAM-2, PAM-4,

PAM-8, PSK-8, PSK-16, QAM-4, QAM-16, QAM-32, QAM-64} for data length of 2000

symbols. When c1, c2, c3 all have variances equal to 0.05, he obtained an average success

rate of about 81% at an SNR of 15 dB, and when the variances were fixed at 0.01, the

average success rate was about 78% at the SNR of 15 dB. When c1, c2, c3 all have variances

equal to 0.1, the average success rate dropped to 75% at the SNR of 15 dB.

As we note from the above, very few works have been reported which address the problem

of modulation classification in multipath scenarios for a large class of constellations. An

example of such a work was reported in [33] where the author considers a 9-class problem

consisting of various linear digital modulations using the rectangular pulse shape. Most of

the classification algorithms for multipath environments use only higher-order statistics [32]

or a combination of both second-order and higher-order statistics [33] for channel estimation

assuming rectangular pulse shape. However, channel estimates obtained using only second-

order statistics usually have much better quality than those obtained using higher-order

statistics. This has motivated our search for a modulation classification algorithm for a

large class of constellations in multipath environments, where the overall impulse response

(inclusive of both the pulse shape and the multipath channel) is estimated using only the

second-order statistics assuming the raised cosine pulse shape which is normally used in a

practical communication system.


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1.2 Problem Statement

We consider a 9-class problem consisting of the following linear digital modulations: {PAM-

2, PAM-4, PAM-8, PSK-8, PSK-16, QAM-4, QAM-16, QAM-32, QAM-64}. The constel-

lations considered are symmetric with zero mean and unit variance. The received signal

y(iTs) is modeled as

y(iTs) =ej2f

oiTs+jo

L1l=0

k=

clskg(iTs kT TT l) + b(iTs) (1.1)

whereTsrepresents the sampling time interval, Trepresents the symbol period,fo(= f

oTs)

represents the normalized carrier frequency offset, o represents the carrier phase offset, sk

represents the symbol sequence,g(.) represents the pulse shape which can be a raised cosine

or a root raised cosine, T

represents the timing offset, L represents the total number of

multipath channel taps, cl represents the multipath tap value of the physical channel, l

represents the delay corresponding to the lth tap, b(.) represents additive white Gaussian

noise of zero mean and variance 2. We assume that an estimate of the noise variance is

available at the receiver and all the constellations are equally likely. The problem is to

estimate the symbol rate and also to identify the modulation format.

1.3 Thesis Organization

The thesis is organized as follows. In Chapter 2, we present the symbol rate estimation algo-

rithm with some simulation results. In Chapter 3, we present the subspace-based method for

blind channel identification, the features used and the algorithm developed for modulation

classification along with simulation results, and Chapter 4 concludes the thesis.


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Chapter 2

Symbol Rate Estimation in

Rayleigh Fading Multipath

Environment in the Presence of

Timing and Carrier Frequency

Offsets

In this chapter, we present a method based on the cyclic autocorrelation function for blindestimation of symbol rate from the received data samples [1]. We assume that a coarse

estimate of the symbol rate is available a priori. This assumption, however, is not a prereq-

uisite for the method. In the absence of any knowledge on the coarse estimate, the method

is still applicable except that computational complexity increases.

In Section 2.1, we present some basic concepts of cyclostationarity. In Section 2.2, we

show the cyclostationarity structure in linear digitally modulated signals. In Section 2.3,

we discuss the effects of channel, carrier and timing offsets on the cyclic autocorrelation

function of the transmitted linear digitally modulated signal. In Section 2.4, we present the

13


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CHAPTER 2. SYMBOL RATE ESTIMATION 14

algorithm for blind estimation of the symbol rate along with some simulation results, and

finally in Section 2.5, we give some concluding remarks.

2.1 Basics of Cyclostationarity

Consider a continuous-time stochastic process x(t). The process x(t) is said to be Nthorder

cyclostationary in the strict sense [36], if its Nth order distribution function

Fx(t)x(t+1)...x(t+N1)(1, 2, . . . , N) =P[x(t) 1, x(t+ 1) 2, . . . , x(t+ N1) N] (2.1)

is periodic in t with some period, say T. That is

Fx(t)x(t+1)...x(t+N1)(1, 2, . . . , N) = Fx(t+T)x(t+1+T)...x(t+N1+T)(1, 2, . . . , N) (2.2)

t R,(1, 2, . . . , N1) RN1 and (1, 2, . . . , N) RN.

The processx(t) is said to besecond-order cyclostationary in the wide-senseif its mean

mx(t) =E[x(t)]1 and autocorrelation functionRx(t+, t) =E[x(t+)x

(t)] are periodic

functions of time t with some period, say T:

mx(t) =mx(t + T) (2.3)

Rx(t+ , t) =Rx(t+ + T, t + T) (2.4)

Since the autocorrelation function is periodic with period T, it can be expressed as a

Fourier series. The Fourier series expansion ofRx(t + , t) is given by

Rx(t + , t) =

n=

Rn/Tx ()ej2(n/T)t (2.5)

where the Fourier coefficients

R

n/T

x () =

1

T

T2

T2

Rx(t + , t) e

j2(n/T)t

dt (2.6)

1E[.] denotes the expectation operation


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are referred to as cyclic autocorrelation functions atcycle frequencies{n/T}nZ.

The cyclic autocorrelation function at the cycle frequency can also be computed from

Rx() = limZ

1

Z

Z2

Z2

Rx(t + , t) ej2tdt (2.7)

We generally use (2.7) for the computation of the cyclic autocorrelation function, since we

do not have a prioriknowledge of the symbol period.

The cyclic spectrum of the signal x(t) at the cycle frequency is defined as the Fourier

transform of the cyclic autocorrelation function at the cycle frequency .

Sx

(f) =

Rx

()ej2fd (2.8)

The above relation is also referred to as the Cyclic Wiener-Khinchin Theorem [36].

2.2 Cyclostationarity in Linear Digitally Modulated Signals

Any linear digitally modulated signal, such as PAM, QAM, PSK, can be represented in the

complex baseband form as follows

x(t) =

k=

skg(t kT) (2.9)

where sk is the information symbol from an unknown signal constellation, T denotes the

symbol period andg(t) represents the pulse shaping function, which in practice has a raised

cosine spectrum. The symbols sk may be real as in the case of PAM or complex as in the

case of PSK and QAM. We can also express (2.9) as

x(t + nT) =

k=

snkg(t+ kT)


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We now present some theorems which illustrate the presence of cyclostationarity in linear

digitally modulated signals.

Theorem 2.2.1. Any linear digitally modulated signal of the form given by (2.9) is second-

order cyclostationary in the wide-sense [1].

Proof:

The symbol sequence is assumed to have zero mean and unit variance. That is, E[sk] = 0

and E[sksm] =km. Then, the mean ofx(t) is

E[x(t)] = E

k=

skg(t kT)

=

k=

E[sk] g(t kT)

= 0 (2.10)

and the autocorrelation function ofx(t) is

Rx(t+ , t) = E

k=

m=

sksmg (t + kT) g (t mT)

=

k=

g (t + kT) g (t kT) (2.11)

If we now calculate Rx(t + + T, t + T), we have from (2.11)

Rx(t + + T, t + T) =

k=

g (t + kT+ T) g (t kT+ T)

=

k=

g (t + kT) g (t kT)

= Rx(t + , t) (2.12)

where we have replaced (k 1) with k in the last step. Thus, the autocorrelation function

and the mean are periodic with a period equal to T (since the mean is zero).


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Theorem 2.2.2. Any linear digitally modulated signal of the form given by (2.9) has three

distinct cycle frequencies at{0, 1T}, when roll-off factor() = 0 [1].

Proof:

From (2.6), the cyclic autocorrelation function of a linear digitally modulated signal at a

cycle frequency kT is given by

RkTx () =

1

T

T2

T2

Rx(t + , t) ej2 k

Ttdt

Using (2.11), we can simplify the above equation as

RkTx () =

1

T

T2

T2

n=

g(t+ nT)g(t nT)ej2kTtdt

= 1T

n=

T2 nTT

2nT

g(t+ )g(t)ej2 kTtdt

= 1

T

g(t+ )g(t)ej2kTtdt

= 1

T

g() g()ej2

kT

(2.13)

where denotes the convolution operation. Taking the Fourier transform of (2.13) on

both sides, we have

SkTx (f) = 1

TG(f)G

f+ k

T

(2.14)

Since the signal is band limited to

(1+)2T ,(1+)2T

with a roll-off factor , it follows from

(2.14) that

SkTx (f) =

= 0 k= {0, 1}0 elsewhere (2.15)

Thus, a linear digitally modulated signal with the raised-cosine or root-raised cosine pulse

shape has only three distinct cycle frequencies at {0, 1T}.


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2.2.1 Sampling of the continous-time signal

Sampling with a rate at least twice the symbol rate yields cyclostationarity. Suppose we

sample the signal at a rate greater than twice the symbol rate. Let this be 1 /Ts2. Choosing

t= iTs and =lTs, the sampled version of (2.7) is given as

R

x (lTs) = limNx

1

2Nx+ 1

Nxi=Nx

Rx(iTs+ lTs, iTs)ej2

iTs (2.16)

Since we only have one realization ofx(iTs), we compute the estimate ofR

x (lTs) as

R

x (lTs) = limNx

1

2Nx+ 1

Nxi=Nx

x(iTs+ lTs)x(iTs)e

j2

iTs (2.17)

As a result of sampling, we see that the cycle frequency at =

k

T in the continous-time casenow becomes kTsT in the discrete-time case. Since the linear digitally modulated signal has

only three distinct cycle frequencies at

0, 1T

, the cycle frequencies of the corresponding

sampled signal are at

0, TsT

. We also note thatR

x (lTs) is periodic in

Tswith a period

equal to 1. This fact suggests that the maximum interval in which the positive non-zero

cycle frequency of the sampled signal can occur is

0, 12

.

2.3 Cyclostationarity in Signals Propagated Through Multi-

path Channels

The received signaly(t) propagated through multipath channel and received in the presence

of carrier frequency offset can be modeled in baseband as

y(t) =ej2f

ot+jo

L1l=0

clx(t l) + b(t) (2.18)

2In practice, an approximate value of the symbol rate is available, and hence, an appropriate value ofTs

can be chosen.


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In the above model, we assume a L-tap multipath physical channel with cl and l rep-

resenting the gain and delay of each of the L paths (we assume 0 0). f

o represents the

carrier frequency offset, o represents the carrier phase offset, x(t) represents the transmit-

ted digital modulated signal and b(t) represents noise. We assume that the noise b(t) is

white Gaussian with zero mean and variance 2.

Theorem 2.3.1. The autocorrelation function of the received signal y(t) is periodic in t

with period equal to the symbol period T and has three distinct cycle frequencies at {0, 1T}

[1].

Proof:

From (2.18), the autocorrelation function of the received signal y(t) is given by

Ry(t+ , t) = E[y(t + )y

(t)]

= E

ej2f

oL1l=0

L1m=0

clcmx(t + l)x

(t m)

+ E[b(t+ )b(t)]

= ej2f

oL1l=0

L1m=0

clcmRx(t + l, t m) +

2() (2.19)

As Rx(t+, t) is periodic in t with period equal to the symbol period T (see (2.12)), the

autocorrelation function of y(t) is also a periodic function in t with period equal to T.SinceRy(t + , t) is a weighted representation ofRx(t + , t), we conclude that y(t) has the

same cycle frequencies as x(t), i.e., three distinct cycle frequencies at {0, 1T}.

2.3.1 Cyclic autocorrelation function of the sampled received signal

Combining (2.9) in (2.18), we have

y(t) =ej2f

ot+j0

L1

l=0

k=

clskg(t kT l) + b(t) (2.20)


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Sampling y (t) at 1/Ts, we obtain

y(iTs) = ej2f

oiTs+jo

L1l=0

k=

clskg(iTs l TT kT) + b(iTs)

= ej2foi+joL1

l=0

k=

clskg(iTs l TT kT) + b(iTs) (2.21)

where T denotes the timing offset and fo(= f

oTs) represents the normalized carrier fre-

quency offset. Now, the autocorrelation function ofy (iTs) is given by

Ry(iTs+ lTs, iTs) = E[y(iTs+ lTs)y(iTs)]

Substituting (2.21) in the above expression and simplifying, we obtain

Ry(iTs+ lTs, iTs) = ej2fol

L

1q=0

L

1p=0

k=

cqc

pg(iTs+ lTs q TT kT)g(iTs p TT kT)

+ 2(lTs)

Combining the terms with respect to the sum over k and using (2.11), we get

Ry(iTs+ lTs, iTs) = ej2fol

L1q=0

L1p=0

cqc

pRx(iTs+ lTs q TT,iTs p TT) + 2(lTs)

From (2.16), the cyclic autocorrelation function ofy(iTs) at the cycle frequency kTs

T

is then

given by

RkTsT

y (lTs) = limNr

1

2Nr+ 1

Nri=Nr

Ry(iTs+ lTs, iTs)ej2 kTs

T i

= ej2folL1q=0

L1p=0

cqcpR

kTsT

x (lTs q+ p) (2.22)

Since the additive noise is stationary, it does not have any non-zero cyclic frequencies.

We note the following from the above steps. The cyclic autocorrelation function of the

received signal is independent of the timing offsetT, noise and the carrier phase offset o. If


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we take the magnitude of the cyclic autocorrelation function of the received signal for a given

lagl, we can eliminate the effect of frequency offset. In the following section, we present the

algorithm developed for blind symbol rate estimation along with some simulation results.

2.4 Symbol Rate Estimation Algorithm and Simulation Re-

sults

2.4.1 Algorithm for symbol rate estimation

Recall from Section 2.2.1 that the cycle frequencies of the sampled signal are at {0, TsT}.

Let the cycle frequency corresponding to TsT be represented by 0. For a finite length

realization of the received sequence, we estimate the cyclic autocorrelation function of the

received discrete-time signal as

RkTsT

y (lTs) = 1

Ny

Ny1i=0

y(iTs+ lTs)y(iTs)e

j2 kTsT i (2.23)

where Ny is the total number samples of the received signal considered in symbol rate

estimation. The algorithm for symbol rate estimation is outlined below.

1. Given the coarse value of the symbol period, say T

, we choose Ts = T

Mswhere Ms is

larger than 2. We then fix the search interval for0 as TsT

+2Ts

, TsT

2Ts . The search

interval depends on the closeness of the coarse estimate to the true value. If the coarse

estimate is poor, we can increase the search interval on both ends. When no coarse

estimate is available, the search interval can be chosen as

0, 12

. We may, however,

point out here that when roll-off factor is less than 0.2, the algorithm becomes sensitive

to noise while searching in larger intervals.

2. We calculate the value of |Ry (lTs)|2 for every (= kTsT ) in the chosen interval for

some lag l (selection of this value is explained in the following section). The value of

which maximises |R

y (lTs)|2

is taken as 0.

3. We then evaluate the symbol rate as 0Ts .


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Table 2.1: Multipath physical channel profile

Tap Variance

c0 0.4078

c1 0.3894

c2 0.1176

c3 0.0833c4 0.0014

2.4.2 Simulation setup

We used 4000 symbols for symbol rate estimation, choosing the pulse shape of a root raised

cosine spectrum with 0.35. The value of the other parameters are chosen as follows:

fo and T are chosen randomly from the intervals [0.2, 0.2] and [0.5, 0.5) respectively.

We consider a 5-tap Rayleigh fading physical channel with the variances of the coefficientsas given in Table 2.1. We consider both uniformly spaced3 and non uniformly spaced4

multipath channels. The other parameter o is not considered in the simulations since

it can be included as a phase component in the multipath channel coefficients. We first

generated the oversampled transmitted signal x(iTs) as

x(iTs) =

Lgk=Lg

skg(iTs kT TT)

where Lg is chosen as 2 and 3 for = 0.5 and 0.35, respectively, and Ts = TP is the

oversampling interval where P may be an integer or a quotient of integers5. We then

generated the oversampled received signal y (iTs) as

y(iTs) =ej2foi

4l=0

clx(iTs l) + b(iTs) (2.24)

3c(t) = c0(t) + c1(t 0.4T) + c2(t 0.8T) + c3(t 1.2T) + c4(t 1.6T)

4c(t) = c0(t) + c1(t 0.4T) + c2(t 0.8T) + c3(t 1.6T) + c4(t 2T)

5If the value of oversampling factor (P = TTs

) is not an integer, we first generate the signal with an

oversampling factor of Np and then decimate the oversampled signal by a factor of Dp to generate thetransmitted signal x(iTs), where P=

Np

Dp.


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We computed the sample variance of the first term on the RHS of (2.24) and generated

noise sequence of appropriate variance to yield a specified SNR.

2.4.3 Selection of the lag parameter

In the above algorithm, the value of the lag-parameter ( l) is to be chosen for the computationof |Ry (lTs)|

2. We now assess the effect of lag (l) on the symbol rate estimation. For

simulations, we chose a QAM-16 symbol set, a root raised cosine pulse shape with = 0.5,

oversampling factorP= 10 and a non-uniformly spaced multipath channel. We assume the

coarse estimate of the symbol period as T

= 8.5Ts, and the search interval for 0 is fixed

as 110.5 ,

16.5

. The SNR is fixed at 5 dB. The increment for () is chosen as 104. We

choose a single value of lag l and compute the which maximises |Ry (lTs)|2 and take this

as 0. The symbol rate estimate is then taken as 0Ts

. If the absolute difference between

the estimated and the actual symbol rates is less than 0.01% of the actual symbol rate,

i.e.0Ts 1T

110000 1T, we declare the estimate as correct. We carried out 1000 trialskeeping the same value of lag and choosing a different realization of symbol sequence, noise

sequence, multipath tap values, frequency offset and timing offset. The ratio of number of

trials in which we declared the estimate as correct to the total number of trials is taken as

the percentage of success. This is repeated for different values of l. Figure 2.1 gives the

success rate for = 0.5 for different lag values l. The above experiment is repeated for

= 0.35 and the corresponding results of percentage of success are given in Figure 2.2. We

note from Figures 2.1 and 2.2 that smaller values of the lag l yield better results compared

to larger values of lags. We therefore, chose the value of lag l = 0 for symbol rate estimation.

2.4.4 Simulation results

We evaluated the performance of the algorithm for two different values of P choosing

uniformly spaced and non-uniformly spaced multipath channels. We chose the values of the

Pas 10 and 12.5. In the case ofP= 10, we assume the approximate value of the symbol

period to be 8.5Ts, and the corresponding search interval of is chosen as 110.5 ,

16.5

. In


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30 20 10 0 10 20 300

10

20

30

40

50

60

70

80

90

100

Lag l

Percentageofsuccessfulestimation

Figure 2.1: Percentage of successful symbol rate estimation for various lag values ( = 0.5, pulse

shape = root raised cosine)

the case ofP= 12.5, we assume the approximate value of the symbol period to be 14 Ts,

and the corresponding search interval of is chosen as116 ,

112

. In both the cases the

increment in () is chosen to be 104. The results are obtained with 10,000 trials for

each modulation type. Table 2.2 gives the percentage of successful estimation of the symbol

rate for non-uniformly spaced channel at an SNR of 5 dB, and Table 2.3 gives for uniformly

spaced channels. Table 2.4 gives the percentage of successful estimation of the symbol rate

for non-uniformly spaced channel at an SNR of 0 dB, and Table 2.5 gives for uniformly

spaced channels. We note that for = 0.5, the percentage of successful estimation is about

98.5% and 95% for both uniformly spaced and non-uniformly spaced channels at SNRs of 5

dB and 0 dB, respectively, while for = 0.35, it falls to about 96% and 88%. These results

convey that the proposed method is relatively insensitive to the value of P as well as the

spacing between the multipath taps.


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30 20 10 0 10 20 300

10

20

30

40

50

60

70

80

90

100

Lag l

Percentageofsuccessfulestimation

Figure 2.2: Percentage of successful symbol rate estimation for various lag values (= 0.35, pulse

shape = root raised cosine)

2.5 Conclusion

In this chapter, we presented a method for blind estimation of the symbol rate for a linear

digitally modulated signal propagated through the multipath channels in the presence of

frequency offset and timing offset. The proposed method shows excellent performance even

at an SNR of 0 dB. When the approximate value of the symbol rate is poor, the size

of the corresponding search interval for 0 needs to be increased which results in more

computations.


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Chapter 3

Blind Modulation Classification in

Rayleigh Fading Multipath

Environment in the Presence of

Timing Offset

In this chapter, we present an algorithm for blind modulation classification in multipath

environment in the presence of timing offset, assuming that the symbol period is known

(which is estimated using the mathod in Chapter 2). We divide the problem of modulation

classification into two steps: We first estimate the composite impulse response of the physical

channel and the pulse shaping function, and then use the estimated channel to compute

various features of interest in order to perform modulation classification.

Algorithms based on higher-order statistics (HOS) for blind channel identification have

been discussed in [8]. However, the channel estimates obtained from HOS tend to have

larger variance compared to those based on lower-order statistics. When a channel is driven

by a stationary process, the second-order statistics (SOS) of the channel output do not

contain phase information, and hence, only minimum phase channels can be identified from

28


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CHAPTER 3. CLASSIFICATION IN MULTIPATH ENVIRONMENT 29

such output. On the other hand, the oversampled channel output, when driven by a cyclo-

stationary process (which is the case with oversampled linear digitally modulated signals),

is a second-order vector stationary process which contains both amplitude and phase in-

formation. Oversampled linear digital communication signals exhibit cyclostationarity, and

exploiting this fact, SOS based blind channel identification was first proposed in [9]-[10].

This approach was later reformulated as a subspace based approach in [12]. In our work,

we use the method proposed in [12] for blind channel identification. We use a combina-

tion of higher-order cumulants and moments as our features of interest in our feature-based

modulation classification system.

In Section 3.1, we present the subspace-based approach for blind channel identification.

In Section 3.2, we briefly introduce the cumulants and moments, which are used in our work,

and describe briefly how they are estimated from the received data using the knowledge of

the estimated channel. In Section 3.3, we present the algorithm developed for hierarchical

modulation classification. In Section 3.4, we provide simulation results to illustrate the

performance of our approach in both multipath fading and flat fading channels in the

presence of timing offset. Section 3.5 gives concluding remarks.

3.1 Subspace Based Method for Channel Identification

The transmitted complex baseband signal is given by

x(t) =

k=

skg(t kT)

where sk is the information symbol from an unknown signal constellation, T denotes

the symbol period and g(t) represents the pulse shaping function, which in practice has

a raised cosine spectrum. Let c(t) denote the baseband equivalent impulse response of

quasi-static multipath fading physical channel and we assume that it is of finite duration.

Let h(t) =g(t) c(t) represent the composite channel impulse response, where denotes


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convolution operation. Then, the received complex baseband signal is represented as

y(t) =

k=

skh(t kT) + b(t) (3.1)

where b(.) is the complex white Gaussian noise with zero mean and variance 2, and it is

assumed to be independent of the signal. We assume that the symbols are uncorrelated and

drawn from a constellation of zero mean and unit variance, i.e., E[sk] = 0 andE[sksl ] =kl,

where kl is the Kronecker delta function.

The development b elow is heavily based on [12]. The received signal is oversampled

with a sampling interval Ts, and a set of Psequences is constructed where P = TTs

. Let

y(i)n = y(t0+ iTs+ nT) denote the i

th sequence where t0 is initial sampling instant which

is modeled as a random variable uniformly distributed in

T2 ,

T2

. Assuming that the

composite impulse response length spans (M+ 1) symbol intervals, we can expressy(i)n as

y(i)n =

Mk=0

snkh(t0+ iTs+ kT) + b(i)n , i= 0, 1, , P 1 (3.2)

where b(i)n =b(t0+ iTs+ nT).

Denoting h(i) [h(t0+iTs), h(t0+iTs+T), , h(t0+iTs+M T)]T, where (.)T denotes the

transpose of (.), we note that sequencey(i)n depends onh

(i). Denotingh(t0+iTs+kT) =h(i)k ,

we can express h(i) = [h(i)0 , h

(i)1 , , h

(i)M]

T. Note that each of h(i), i = 0, 1, , P 1,

is a symbol spaced channel. Stacking N successive samples of y(i)n and denoting y

(i)n =

[y(i)n , y

(i)n1, , y

(i)nN+1]

T, we have (Ndenotes the temporal window of length > M)

y(i)n = H(i)Nsn+ b

(i)n , i= 0, 1, , P 1 (3.3)

whereb(i)n = [b

(i)n , b

(i)n1, , b

(i)nN+1]

T is aN 1 vector, sn = [sn, sn1, , snNM+1]T is

a (N+ M)1 vector and H(i)Nrepresents the filtering matrix of size N(N+ M) associated

with h(i) and is defined as


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H(i)N =

h(i)0 h

(i)M 0 0

0 h(i)0 h

(i)M 0 0

... ...

0 h(i)0 h

(i)M

(3.4)

Since we have a total of P sequences ofy(i)n for i= 0, 1, , P 1, we can represent the

entire system consisting of the Psequences of (3.3) as given below.

y(0)n

...

y(P1)n

=

H(0)N...

H(P1)N

sn+

b(0)n

...

b(P1)n

(3.5)

Denoting yn = [y

(0)T

n , y

(1)T

n , , y

(P1)T

n ]

T

, HN = [H

(0)T

N , H

(1)T

N , , H

(P1)T

N ]

T

and bn =[b

(0)Tn , , b

(P1)Tn ]T, we can rewrite (3.5) as

yn= HNsn+ bn (3.6)

We now outline the subspace based method for estimating the P(M+ 1) coefficients

h= [h(0)T, h(1)T, , h(P1)T]T (3.7)

from the set of observations yn. The steps given here are adopted from [12]. Note that

h(i) refers to the discrete-time impulse response of the ith symbol spaced channel. We first

compute the data covariance matrix Ry (of size P N P N) ofyn.

Ry = Eyny

Hn

(3.8)

= HNRsHHN+ Rb (3.9)

where Rs

= Esn

sHn represents the covariance matrix (of size (N +M) (N +M))

of the transmitted signal, Rb = E

bnbHn

represents the noise covariance matrix of size


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P N P N, and (.)H denotes the conjugate transpose of (.). We assume thatRs is full rank

and Rb= 2I. We have the following from [12].

Theorem 3.1.1. The MatrixHN is full column rank, i.e., rank (HN) =M+ N, if

1. the polynomialsh(i)(z) Mj=0 h(i)j zj , i= 0, 1, , P 1, have no common zeros,2. N is greater than the maximum degreeM of the polynomials, i.e., N > M, and

3. at least one polynomialh(i)(z) has degreeM.

Let0 1 PN1 denote the eigenvalues ofRy. Since Rs is full-rank, the signal

partof the covariance matrix Ry, i.e., HNRsHHN, has rank M+ N. Hence,

i > 2 i= 0, , M+ N 1

i = 2 i= M+ N, , P N 1 (3.10)

Denote the eigenvectors corresponding to {0, 1, M+N1} by {d0, d1, , dM+N1}

and the remaining eigenvectors corresponding to {M+N, M+N+1, PN1} byg0, g1, , gPNMN1

. Defining

D = [d0, d1, , dM+N1] P N (M+ N)

G = [g0, g1, , gPNMN1] P N (P N M N), (3.11)

we can express Ry as

Ry =D diag(0, , M+N1) DH + 2GGH. (3.12)

The columns ofD span the signal subspaceof dimension (M+N), while those ofG span

the noise subspace. The signal subspace is also the linear space spanned by the columns of

HN. From the orthogonality between the noise and signal subspaces, the columns ofHN


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are orthogonal to any vector in the noise subspace. Hence, we have

gHi HN= 0 0 i < PN M N (3.13)

In practice, we only have finite data to work with. We estimate the data covariance matrix

as

Ry = 1

N

N1n=0

ynyHn (3.14)

where N is the total number of symbols considered in the channel identification and classi-

fication. The eigendecomposition of (3.14) gives sample estimates of the noise eigenvectors,

{gi}. We therefore solve for HNby minimizing the following quadratic form

q(h) =

PNMN1

i=0

|gHi HN|2 (3.15)

In [12], it is shown that minimization of the quadratic form (3.14) is equivalent to

minimizing the following quadratic form

q(h) =hHQh with Q=

PNMN1i=0

GiGHi . (3.16)

where Gi is theP(M+ 1) (M+ N) filtering matrix associated withgi, and it is generated

as follows [12].

Let the P N1 vectorgibe represented asgi= [g(0)Ti , , g

(P1)Ti ]

T, whereg(0)i , , g

(P1)i

are the N 1 vectors picked from gi as follows. Letgi = [gi,0, gi,1, , gi,PN1]T.

Then, g(k)i = [gi,kN, gi,kN+1, , gi,kN+N1]

T. Now the (M+ 1) (M+N) filtering


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matrix G(l)i associated withg

(l)i is given by

G(l)i =

gi,lN gi,lN+1 gi,lN+N1 0 0

0 gi,lN gi,lN+1 gi,lN+N1 0

0 0 gi,lN gi,lN+1 gi,lN+N1

(3.17)

TheP(M+1)(M+N) filtering matrix Gi, associated withgi, is obtained by stacking

Pfiltering matrices as given below.

Gi = [G(0)Ti ,

G(1)Ti , ,

G(P1)Ti ]

T (3.18)

The solution vector which minimizes q(h) subject to||h|| = 1 is the unit-norm eigenvector

associated with the smallest eigenvalue ofQ.

3.1.1 Non-identificable channels

Tugnait [11] has shown that the discrete-time multipath physical channels with delays equal-

ing integer multiples ofT are not identifiable from the second-order cyclostationary statis-

tics. Also, those with delays equaling integer multiples of T /2 are not identifiable when

the oversampling factor is an even integer. In view of this, we used physical channelc(t)

whose discrete-time impulse response does not fall under the above channel classes in our

classification problem.

From the estimated channel h, we choose h(k)

having the maximum norm and use it in

our classification problem. For notational convenience later, we denote the maximum-norm

component of the composite impulse response estimated from Ry as a(0), a(1), , a(M),

and the corresponding sequence of the received signal as f(n), n= 0, 1, ,N 1. That is,

a(i) =h(k)i with

h(k)i denoting thei

th coefficient ofh(k)

, andf(n) =y(k)n , n= 0, 1, ,N 1.


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Note that the impulse response coefficients {a(i)} and the data sequence {f(n)} are symbol

spaced. In the following sections, we discuss the features and the algorithm used in the

hierarchical modulation classification.

3.2 Features Used in the Classification

In view of the representation of the maximum norm component ofhas a(i) and the corre-

sponding data sequence as f(n), and combining with (3.2), we have

f(n) =

Mi=0

a(i)sni+ w(n) (3.19)

wherew(n) =b(k)n . We assume that an estimate of noise variance is available at the receiver

and that all the constellations are equally probable.

3.2.1 Definitions of the features used

In our algorithm, we used second-, fourth- and sixth-order cumulants and eighth-order

moment. These features, as applied to the symbols sn, are defined as [8]

C20,sn = E[s2n] (3.20)

C21,sn = E[|sn|2] (3.21)

C40,sn = E[s4n] 3C

220,sn (3.22)

C42,sn = E[|sn|4] |C20,sn |

2 2C221,sn (3.23)

M41,sn = E[s3ns

n] (3.24)

M42,sn = E[|sn|4] (3.25)

C63,sn = E[|sn|6] 6|C20,sn ||M41,sn | 9C21,snM42,sn+ 18|C20,sn |

2C21,sn+ 12C321,sn(3.26)

M80,sn = E[s8n] (3.27)

where the super script denotes complex conjugate. The values of the above features for

the 9 underlying modulations (assuming unit power constellations) are given in Table 3.1.


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features in our algorithm.

Since, in practice, we only have finite data, we replace the features off(n) in the above

expressions with their finite data estimates, and use the resulting estimates of the features

ofsn in our algorithm. For example, the finite data estimate of |C40,sn |, denoted by |C40,sn |,

is given by

|C40,sn | =

M

i=0 |a2(i)|2

Mi=0 a

4(i)

C40,f(n)

(C21,f(n) 2)2

(3.33)

where C40,f(n)and C21,f(n)are the finite data estimates ofC40,f(n)and C21,f(n), respectively.

Similarly, we obtain the estimates of the features in (3.28) and (3.30) to (3.32). We may

mention here that the asymptotic values of the finite data estimates ofsnobtained as above

tend to those given in Table 3.1.

In our algorithm we used the distance metric approach. In this approach, the metric

used for classification is the Euclidean distance between the estimated and asymptotic values

of an individual feature. For example, consider a problem where we have to identify the

constellation of a received signal amongn hypotheses corresponding to the n constellations.

Let the feature be X, its estimate obtained be Xand its asymptotic value under the mth

hypothesis be Xm. We then evaluate dm for the nconstellations

dm = |X Xm|, m = 1, 2, , n (3.34)

and decide in favor ofith constellation ifdm is minimum for m= i.

3.3 Algorithm for Hierarchical Blind Modulation Classifica-

tion

The algorithm, shown in Figure 3.1, uses hierarchical classification of the underlying mod-

ulations. Observe from the figure that the classification tree involves several stages. We

briefly describe each stage. The threshold values used in the classification are determined

from Table 3.1.


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Figure 3.1: Algorithm for hierarchical modulation classification (modulations considered are PAM,PSK and QAM constellations)

STAGE-1

In this stage, we determine whether the received signal constellation belongs to the

PAM subclass or the PSK/QAM subclass. We use |C20,sn | as the feature and fix the

threshold value at 0.5.

STAGE-2

In this stage, we seperate PAM-2 from the PAM subclass, if the incoming signal

is classified as PAM subclass in the first stage, or we separate the PSK and QAM


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subclasses. In the former case, we use C42,sn as the feature and fix the threshold value

at1.68. In the latter case, we use the feature |C40,sn |

C42,snand fix the threshold value at

-0.1377.

STAGE-3

In this stage, we separate PAM-4 from PAM-8 if the second stage decides in favor

of the {PAM-4, PAM-8} subclass, or separate the QAM subclass into QAM-32 and

QAM-{4, 16, 64} subclass if the second stage decides in favor of the QAM subclass

or we separate PSK-8 from PSK-16. To separate PAM-4 from PAM-8, we use C63,sn

as the feature with the threshold value as 7.75445. To separate PSK-8 from PSK-16,

we use | M80,sn | with the threshold value as 0.5. To separate QAM-32 from{QAM-

4, QAM-16, QAM-32, QAM-64} subclass, we use |C40,sn | as the feature setting the

threshold value at 0.4045.

STAGE-4

In this stage, we seperate QAM-4 from {QAM-16, QAM-64} subclass, if the third

stage decides in favor of the {QAM-4, QAM-16, QAM-64}subclass. We use C42,sn as

the feature with the threshold value set at 0.84.

STAGE-5

In this stage, we separate QAM-16 from QAM-64, if the fourth stage decides in favor

of the QAM-{16, 64} subclass. We use C63,sn as the feature and set the threshold

value at 1.9386.

3.4 Simulation Results

We have conducted extensive simulations to study the quality of the channel estimate, and

the classification performance obtained with the maximum-norm component h(k)

of the

overall composite channel impulse response estimateh. In our simulations, the oversampled

factor P, where P = TTs , is chosen as 4 for all the constellations considered. The pulse


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shaping function g(t) has raised cosine spectrum with roll-off of 0.5. Restricting the pulse

shape to2T to 2T, we first generated the oversampled transmitted signal x(iTs) as

x(iTs) =2

k=2

skg(iTs kT TT)

where T is a random variable chosen from the interval [-0.5, 0.5). We assumed a 5-tap

Rayleigh fading physical channel of duration 2Tas given below

c(nTs) =c0(nTs) + c1(nTs 2Ts) + c2(nTs 3Ts) + c3(nTs 6Ts) + c4(nTs 8Ts) (3.35)

The coefficients cis are complex Gaussian random variables with variances as given in

Table 2.1. We then generated sampled received signaly (iTs) as

y(iTs) =

4l=0

clx(iTs l) + b(iTs) (3.36)

where l assumes the values given in (3.35), and b(iTs) is zero mean white Gaussian noise.

We computed sample variance of the first term on the RHS of (3.36) and generated noise

sequence of appropriate variance to yield a specified SNR.

For all the considered constellations, we chose the length of the temporal window, N,

as 10 and number of symbols, N, as 2000 or 4000, and estimated the composite channel

following the method described in Section 3.1. Selecting the maximum-norm component ofthe estimated channel and the corresponding received sequence, we computed the estimates

of the features and performed classification as per the algorithm shown in Figure 3.1. For

each constellation, we conducted 10,000 Monte Carlo runs of the above experiment and

determined the percentage of correct classification. This is repeated for several values of

SNR for each constellation. Tables 3.2 and 3.3 give the percentage of correct classification for

2000 and 4000 symbols, respectively. The tables also give the average percentage of correct

classification for each SNR, evaluated by averaging over all the considered constellations (see

the last row). Note that the classification performance is poor in the cases of PAM-8 and


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15 16 17 18 19 20 21 22 23 24 250

0.5

1

1.5

2

2.5

3

SNR (dB)

2000 symbols

4000 symbols

Figure 3.2: Quality of the estimated channel in Rayleigh fading multipath physical channel case

3.4.1 Flat fading Rayleigh channel

In [31], the authors considered a flat fading Rayleigh channel and studied the classification

performance for the 9-class problem same as the one considered here, assuming perfect

knowledge of the channel. They used eighth-order cyclic cumulants and distance metric.

But, unlike in our case, the metric used in [31] is based on a vector of features corresponding

to all the considered constellations. To see how our method performs compared to their

approach, we considered a flat fading Rayleigh physical channel choosing the variance of

the single coefficient as unity, and evaluated the classification performance. The results

are given in Table 3.6. Unlike in the earlier simulations, we performed these simulations

for SNR values of 5, 10 and 15 dB so that a comparison with the results of [31] can be

made. With our approach, the values of average percentage of correct classification are68.98, 88.93 and 98.20 while the corresponding values in [31] are approximately 63, 81 and


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Table 3.6: Percentage of correct classification in Rayleigh fading flat physical channel (fadingcoefficient of unit variance, N= 4000 and 10,000 trials)

SNR (dB) 5 10 15

PAM-2 99.99 100.00 100.00PAM-4 72.54 97.54 99.82

PAM-8 20.24 52.93 92.55

PSK-8 97.69 100.00 100.00

PSK-16 41.49 100.00 100.00

QAM-4 100.00 100.00 100.00

QAM-16 71.14 97.38 99.81

QAM-32 97.17 99.99 100.00

QAM-64 20.56 52.49 91.62

Average 68.98 88.93 98.20


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Chapter 4

Conclusion

In this thesis, we addressed the problem of blind symbol rate estimation and modulation

classification for a 9-class problem in Rayleigh fading multipath environment. We developed

an algorithm for symbol rate estimation, which performs extremely well at low SNRs in

the presence of multipath, timing and carrier frequency offsets. We have also developed

an algorithm for modulation classification in multipath scenarios in the presence of timing

offset. In our classification algorithm, the overall impulse response is estimated using SOS,

and using the estimated channel we computed the features of interest to perform modulation

classification. We obtained very good results in both the multipath and the flat fading

scenarios. The proposed algorithm gives better results in flat fading environment than [31]1

even thougha prioriknowledge of the pulse shape coefficients and channel gain are assumed

in [31].

4.1 Future Work

Causey and Barry [35] showed that second-order statistics based identification is possible

only with the knowledge of the carrier frequency offset. One can investigate the problem of

classification in multipath environment in the presence of both timing and carrier frequency

offsets. Also, the modulation formats considered in this thesis are memoryless linear digital1[31] gives the results for flat fading scenario only

46


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CHAPTER 4. CONCLUSION 47

modulations. The problem becomes more complex when the class size is increased and when

the set of modulations consists of b oth with memory and without memory. Classification

problem will be a challenging task in such cases.


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Documents

Anantha Krishna Karthik-Thesis