hierarchical blind modulation classification in the...

HIERARCHICAL BLIND MODULATION CLASSIFICATION IN

THE PRESENCE OF CARRIER FREQUENCY OFFSET

A thesis submitted in partial fulfilment of

the requirements for the degree of

Master of Science (by research)

in

Communication Systems and Signal Processing

by

Chaithanya Velampalli

200736002

[email protected]

Communications Research Center

INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY

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

May 2010

INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY

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

CERTIFICATE

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

Classification in the Presence of Carrier Frequency Offset ” by Chaithanya Velampalli,

has been carried out under my supervision and it is fully adequate in scope and quality

as a dissertation for the degree of Master of Science.

Date Prof V.U. Reddy (Advisor)

Abstract

Blind modulation classification deals with identification of modulation formats from

received signal without the knowledge of the type of modulation transmitted. The

problem becomes more challenging in real world scenarios when there are synchro-

nization errors such as frequency offset and timing offset, and multi-path fading. We

consider the problem of modulation classification in the presence of carrier frequency

offset.

Several algorithms, based on higher-order cumulants of differentially processed re-

ceived signal and cyclic cumulants of the received signal, have been recently proposed

for blind classification in the presence of frequency offset. Since the variance of the

estimates of these features is high, more data and/or high SNR is required for good

classification performance.

We consider 9-class and 10-class problems and propose hierarchical classification

using a combination of moments of the received and differentially processed received

signal, unlike in previous work where only the cumulants of the differentially processed

received signal are used. The motivation for our approach comes from the fact that for

a given data size, the estimates of the moments of the received signal have less variance

compared to the cumulants of differentially processed received signal. Simulations are

used to illustrate the performance of the proposed algorithm.

iv

Acknowledgement

First and foremost, I would like to express my gratitude to my advisor, Prof. V. U.

Reddy for his support and guidance. I am grateful to him for giving me this opportu-

nity to explore my interests and pursue research in the field of Communications and

Signal Processing.

I am thankful to my friends and labmates at CRC who made my stay at IIIT-H

memorable.

Last, but not the least, I want to thank my parents for their moral support and

for encouraging me to take up studies in my field of interest.

v

Contents

Abstract iv

Acknowledgement v

1 Introduction 1

1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Features Used for Classification 6

2.1 Moments of Received Signal . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Differential Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Algorithm for Hierarchical Classification 12

3.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Results 15

4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 10-Class Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Conclusion 20

5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

vi

A Evaluating Feature Used for 4-PSK & O-QPSK Classification 21

B Derivation of Moments of Differentially Processed Signal 26

Bibliography 30

vii

List of Tables

2.1 M42,x(n) for various considered constellations (x(n) is of unit variance ) 8

2.2 Moments of X2(n) for various considered constellations (x(n) is of unit

variance ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 TX1(n) for 4-PSK and O-QPSK . . . . . . . . . . . . . . . . . . . . . . 11

4.1 Performance of the proposed algorithm as percentage of correct classification

(1000 symbols with two samples per symbol and ε=0.01 are used in 105 trials) 16

4.2 Moments of x(n) and X2(n) for 8- & 16-PSK ( x(n) is of unit variance ) 17

4.3 Performance of the proposed algorithm as percentage of correct classification

for 10-class problem (1000 symbols with two samples per symbol and ε=0.01

are used in 105 trials) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A.1 Symbol transitions along with respective probabilities for 4-PSK . . 21

A.2 X1(n) and X1(n− 1) along with respective probabilities for 4-PSK . 22

A.3 Absolute difference along with respective probabilities for 4-PSK . . 22

A.4 Symbol transitions along with respective probabilities for 4-PSK . . 22

A.5 X1(n) and X1(n− 1) along with respective probabilities for 4-PSK . 23

A.6 Absolute difference along with respective probabilities for 4-PSK . . 23

A.7 Symbol transitions along with respective probabilities for O-QPSK . 24

A.8 X1(n) and X1(n− 1) along with respective probabilities for O-QPSK 24

A.9 Absolute difference along with respective probabilities for O-QPSK . 24

viii

List of Figures

3.1 Algorithm for hierarchical modulation classification using distance met-

ric (modulations considered are PAM, PSK and QAM) . . . . . . . . 13

4.1 Average percentage of correct classification . . . . . . . . . . . . . . . 16

4.2 Algorithm for hierarchical modulation classification using distance met-

ric for 10-Class problem (modulations considered are PAM, PSK and

QAM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

ix

Chapter 1

Introduction

Automatic modulation classification is of major importance in civilian and military

applications. The interest in this area has increased in recent years owing to the

advances in reconfigurable signal processing systems, especially in software defined

radio and cognitive radio.

1.1 Previous Work

The different approaches of blind modulation classification can be broadly classified

into two groups; (1) likelihood based [1, 2] and (2) feature based [3]-[10]. The likeli-

hood based approach treats modulation classification as multiple hypotheses testing

problem and uses maximum-likelihood framework. 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 li-

brary.

Modulation classification in the presence of timing and frequency errors is a diffi-

cult problem. When the local oscillators at transmitter and receiver are not synchro-

nized, a frequency offset arises which results in progressive phase shift in the time

domain data of the received signal at baseband. Blind classification with frequency

1

CHAPTER 1. INTRODUCTION 2

offset is well addressed in recent literature. Two approaches have been followed in this

regard. First approach utilizes cumulants of differentially processed signal. Differen-

tial processing operation converts frequency offset into a fixed phase offset. Second

approach makes use of cyclic cumulants. We review briefly the recent work based on

these two approaches.

Studies reported in [3] to [5] are based on the first approach. In [3], the authors

consider a 4-class problem consisting of BPSK, 4-PAM, 8-PSK and 16-QAM, and

suggest the use of fourth-order cumulant of differentially processed received signal.

A 10-class problem consisting of BPSK, 4-PAM, QPSK, O-QPSK, 8-PSK, 16-PSK,

8-QAM, 32-QAM, 16-QAM, 64-QAM is considered in [4] where second-, fourth- and

eighth-order cumulants of differentially processed received signal are used. Classifi-

cation performance is given for 10 and 15 dB SNR for data sizes varying from 500

to 3000 symbols, with raised cosine pulse shape and 50 samples per symbol. In [5],

the authors use fourth-order cumulants of differentially processed received signal for

classification of QAM constellations corresponding to 16-, 64- and 256-QAM, 32- and

128-QAM (cross), 16- and 32-QAM (star) and 8-QAM. They first classify them into

two broad groups, one consisting of 16-, 64- and 256-QAM square constellations and

the other consisting of the rest, and then perform inter classification in these groups.

Detection performance for SNRs in the range of 5 to 30 dB, with 2000 symbols using

raised cosine pulse shape and 4 samples per symbol, is given in [5].

Studies based on cyclic cumulants are reported in [6] and [7]. In these studies, the

carrier frequency offset is estimated from a preprocessing step, and it is assumed to

be perfect. A hierarchical procedure is suggested for classification in [6] where 2-class

problems consisting of groups i) QPSK and 16-QAM, ii) 4- and 8-PAM, iii) 16- and

64-QAM, are considered separately, using eighth-order cyclic cumulants. Classifica-

tion results are given for data sizes ranging from 1000-7000 symbols at SNR of 5 dB

for the first group, 1000 to 20,000 symbols for SNRs of 5 and 10 dB for the other

two groups, with raised cosine pulse shape. A 2-class problem consisting of 4-QAM

and 16-QAM is considered in [7] where a feature vector comprising of fourth-, sixth-


and eighth-order cyclic cumulant is used. Results of average percentage of correct

classification are provided for SNR of 7 dB for data size of 900 symbols, with raised

cosine pulse shape and 9 samples per symbol. For large class problems, use of hierar-

chical classification is suggested in [7]. A survey of different methods of modulation

classification is given in [11].

1.2 Research Motivation

The problem with the above approaches, in general, is large variance of the estimates

and because of this, more data and high SNR are required to obtain good classifi-

cation performance. This motivated us to look for features that are insensitive to

frequency offset and also, their estimates have lower variance for a given data size.

We propose to use moments of the received and differentially processed received

signal, and classify the modulations hierarchically. We use second-order moments of

differentially processed received signal for broad classification and fourth-order mo-

ments of received and differentially processed received signal for finer classification.

The motivation for using the moments of the received signal is the following: i) The

fourth-order moment is insensitive to the carrier frequency offset and ii) for a given

data size, its variance is smaller than that of features used in [3] to [7].

In cases where the asymptotic values of the fourth-order moment of the received

signal is same or very close to each other for certain modulations, we use fourth-order

moment of differentially processed received signal and other similar feature. Unlike

in [6] and [7], we do not use any preprocessing step to estimate the carrier frequency

offset.


1.3 Signal Model

We work with baseband signal and use rectangular pulse shape with two samples per

symbol x(n). We consider 9-class problem consisting of 2-, 4- and 8-PAM, 4-PSK,

O-QPSK (Offset-QPSK) and 8-PSK, 16-, 32- and 64-QAM, and assume that all the

constellations are symmetric with zero mean and unit variance. We consider additive

white Gaussian channel. The received signal y(n) is modeled as

y(n) =√

Pej(2πεn+θ)x(n) + w(n) (1.1)

where θ is the phase offset caused either due to phase mismatch between the trans-

mitter and receiver oscillators or constellation rotation at the transmitter, ε is the

normalized carrier frequency offset (∆fTs = ε), P is the power in the received signal

and w(n) is a sample function of a circularly symmetric zero mean complex white

Gaussian noise process with variance σ2w which is uncorrelated with x(n). Here, ∆f

and Ts denote carrier frequency offset and spacing between consecutive samples, re-

spectively. We assume that an estimate of noise variance is available at the receiver

(similar assumption is made by others, see [3]). We also assume that all modulations

and constellations are equally likely.

1.4 Contributions

The contributions of this thesis are as follows. We classify large class problems (9-

Class and 10-Class) in the presence of carrier frequency offset. We use a combination

of features derived from received signal and differentially processed received signal.

These features are insensitive to carrier frequency offset. We use distance metric based

approach and the classification is performed in a hierarchical manner with features

of lower order for broad classification and higher order for finer classification.


1.5 Thesis Organization

This thesis is organized as follows. In Chapter 2, we discuss the features used in the

classification. Algorithm for modulation classification is given in Chapter 3. Simu-

lation results of the algorithm are given in Chapter 4. Chapter 5 concludes with a

summary and suggestions for future work.

Chapter 2

Features Used for Classification

Our goal is to look for the features which are insensitive to frequency offset and whose

finite data estimates have lower variance. One such feature is fourth moment of the

received signal. But, this feature alone is not adequate for large class problem. We

therefore derive other features from differentially processed received signal and use

them along with the fourth moment. We first define the required features of trans-

mitted signal x(n) and then derive their relations to those of received signal y(n), as

we have to work with the received signal.

2.1 Moments of Received Signal

The second- and fourth-order moments of x(n) are defined as

M21,x(n) = E[x(n)x∗(n)] = E[|x(n)|2] (2.1)

M42,x(n) = E[x(n)x(n)x∗(n)x∗(n)] = E[|x(n)|4] (2.2)

Combining (1.1) with the above definitions, we have

M21,y(n) = PE[|x(n)|2] + σ2w (2.3)

6

CHAPTER 2. FEATURES USED FOR CLASSIFICATION 7

which gives, since x(n) is of unit variance,

P = M21,y(n) − σ2w (2.4)

Expressing (2.2) for y(n) and combining it with (1.1), we obtain the following

relation

M42,x(n) =M42,y(n) − 4(M21,y(n) − σ2

w)σ2w − 2σ4

w

(M21,y(n) − σ2w)2

(2.5)

where we have used (2.4) for P .

In the case of finite data with N samples, we first estimate the moments by

replacing ensemble averages with time averages. That is, second-order moment of y(n)

is given by M̂21,y(n) = 1N

∑Nn=1 |y(n)|2 and fourth-order moment of y(n) by M̂42,y(n) =

1N

∑Nn=1 |y(n)|4. We then compute the feature in (2.5) with these estimates and refer

to the result as the estimated normalized feature. Thus, the estimated normalized

fourth-order moment of x(n), denoted as M̃42,x(n), is evaluated from the received

signal as

M̃42 =M̂42,y(n) − 4(M̂21,y(n) − σ2

w)σ2w − 2σ4

w

(M̂21,y(n) − σ2w)2

(2.6)

where we have dropped the second subscript x(n) for convenience. Estimate of P is

obtained as

P̂ = M̂21,y(n) − σ2w (2.7)

We may mention here that the asymptotic value of M̃42 is M42,x(n). Table 2.1 gives

M42,x(n) for the underlying constellations.

We note the following from Table 2.1. (i) Asymptotic values of M̃42 are same for

different PSK constellations. Hence, inter-classification among PSK constellations is

not possible using M̃42. (ii) Asymptotic values of M̃42 for 16- and 32 QAM are very

close to each other, and hence, M̃42 is not the appropriate feature for classification be-

tween them with short data size and/or at low SNR. The above observations motivate

us to consider the use of differential processing.


Table 2.1: M42,x(n) for various considered constellations (x(n) is of unit variance )

Constellation M42,x(n)

2-PAM 14-PAM 1.648-PAM 1.76194-PSK 1

O-QPSK 18-PSK 1

16-QAM 1.3232-QAM 1.3164-QAM 1.38

2.2 Differential Processing

Differential processing operation converts normalized frequency offset into fixed phase

offset. We now describe the moments of the differentially processed signal. Let

Y2(n) = y∗(n−2)y(n) denote the differentially processed received signal and similarly

X2(n) = x∗(n− 2)x(n). Combining (1.1) with the above definition, we obtain

Y2(n) = Pej4πεx∗(n− 2)x(n) + (√

Pe−j(2πε(n−2)+θ)x∗(n− 2)

w(n)) +√

Pej(2πεn+θ)x(n)w∗(n− 2) + w∗(n− 2)w(n)

which can be expressed as

Y2(n) = Pej4πεX2(n) +√

Pe−j(2πε(n−2)+θ)x∗(n− 2)w(n)

+√

Pej(2πεn+θ)x(n)w∗(n− 2) + w∗(n− 2)w(n) (2.8)

We now relate the second- and fourth-order moments of X2(n) to those of Y2(n).

From the definition,

M20,Y2(n) = E[Y2(n)Y2(n)] = E[Y 22 (n)]


Substituting (2.8) for Y2(n) and simplifying (see Appendix B for details), we get

M20,X2(n) =e−j8πεM20,Y2(n)

(M21,y(n) − σ2w)2

(2.9)

where the denominator corresponds to P 2. Similarly, we obtain (see Appendix B)

M40,X2(n) =e−j16πεM40,Y2(n)

(M21,y(n) − σ2w)4

(2.10)

The phase offset is removed by choosing the absolute values. In arriving at (2.9)

and (2.10), we have used two assumptions underlying the signal model: a) the noise

and the signal are uncorrelated and b) the noise is white.

The estimated normalized quantities of the moments (2.9) and (2.10) in the case

of finite data, evaluated from the differentially processed received signal, are given by

|M̃20,dp| =|M̂20,Y2(n)|

(M̂21,y(n) − σ2w)2

(2.11)

and

|M̃40,dp| =|M̂40,Y2(n)|

(M̂21,y(n) − σ2w)4

(2.12)

We added a second subscript dp to denote that these features refer to differentially

processed signal. The asymptotic values of |M̃20,dp| and |M̃40,dp| are |M20,X2(n)| and

|M40,X2(n)|, respectively, and Table 2.2 gives these values of X2(n).

We note from Table 2.2 that |M20,X2(n)| can be used for broad classification of

the incoming signal into PAM sub-class and PSK/QAM sub-class, |M40,X2(n)| for in-

ter classification among QAM constellations and, for separating 4-PSK and O-QPSK

from 8-PSK. We still need a feature which can separate 4-PSK from O-QPSK.

Now, consider Y1(n) = y∗(n − 1)y(n) and X1(n) = x∗(n − 1)x(n). It is easy to

obtain the following relation between them.


Pe−j(2πε(n−1)+θ)x∗(n− 1)w(n)


Table 2.2: Moments of X2(n) for various considered constellations (x(n) is of unitvariance )

Constellation |M20,X2(n)| |M40,X2(n)|2-PAM 1 14-PAM 1 2.68968-PAM 1 3.10434-PSK 0 1

O-QPSK 0 18-PSK 0 0

16-QAM 0 0.462432-QAM 0 0.036164-QAM 0 0.3832

+√

Pej(2πεn+θ)x(n)w∗(n− 1) + w∗(n− 1)w(n) (2.13)

Denote the ensemble average of the square of the absolute value of the difference

of adjacent samples of Y1(n) as

TY1(n) = E[|Y1(n)− Y1(n− 1)|2]

and similarly

TX1(n) = E[|X1(n)−X1(n− 1)|2]

Combining the above definitions with (2.13), we obtain (after some manipulations)

TY1(n) = P 2TX1(n) + 4Pσ2w + 2σ4

w

from which we have the following

TX1(n) =TY1(n) − 4(M21,y(n) − σ2

w)σ2w − 2σ4

w

(M21,y(n) − σ2w)2

, (2.14)

where we have used (2.4) for P . The values of TX1(n) for 4-PSK and O-QPSK,

evaluated in Appendix A, are given in Table 2.3. Note that TX1(n) can be used as the

feature for separating 4-PSK from O-QPSK.


Table 2.3: TX1(n) for 4-PSK and O-QPSK

Constellation TX1(n)

4-PSK 2O-QPSK 1

The estimated normalized value of TX1(n), evaluated from Y1(n), is given by

T̃ =T̂Y1(n) − 4(M̂21,y(n) − σ2

w)σ2w − 2σ4

w

(M̂21,y(n) − σ2w)2

(2.15)

We use (2.6), (2.11), (2.12) and (2.15) as features for classification.

Chapter 3

Algorithm for Hierarchical

Classification

3.1 Approach

We use distance metric based approach in the classification algorithm. In this ap-

proach, the metric used for classification is the distance between the estimate and its

asymptotic value of an individual feature. Consider the problem where we have to

identify the constellation of the received signal from among n hypotheses correspond-

ing to n constellations. Let the feature used for this be F, its estimated normalized

value obtained from finite data be F̃, and its asymptotic value under kth hypothesis

be Fk. We then calculate di under each of the n hypotheses

di = |F̃− Fi|2, i = 0, · · · , n− 1. (3.1)

and decide in favor of kth hypothesis if di is found to be minimum for i=k. We perform

the classification in four stages as described below (see Fig. 3.1).

12

CHAPTER 3. ALGORITHM FOR HIERARCHICAL CLASSIFICATION 13

3.2 Algorithm

STAGE-1

In this stage, we decide whether the incoming signal belongs to PAM constellation

set or to PSK/QAM set. From Table 2.2 we note that the asymptotic values of

|M̃20,dp| corresponding to the above two sub-classes are well separated, and hence,

can serve as the feature for separating into these two sub-classes. Let H0 and H1

represent the hypotheses that the incoming signal is from PAM set and PSK/QAM

sets, respectively. We apply the distance metric choosing the threshold as 0.5. From

the received data, we evaluate |M̃20,dp| . If this is greater than or equal to 0.5 we

choose the hypothesis H0, otherwise H1.

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

CHAPTER 3. ALGORITHM FOR HIERARCHICAL CLASSIFICATION 14

STAGE-2

In this stage, we perform inter classification among PAM constellations if the incoming

signal is classified as PAM sub-class in the first stage or separation between PSK and

QAM sub-classes otherwise. Note from Table 2.1 that the asymptotic values of M̃42

for the PSK set is 1 while it is around 1.32 for QAM set. We use distance metric

choosing the threshold as 1.155. Let hypothesis H0 represent PSK set and H1 QAM

set. From the received data, we evaluate M̃42. If this is less than 1.155, we choose

the hypothesis H0, otherwise H1. For inter classification among PAM, we choose the

thresholds as shown in Fig. 3.1 and classify into 2-, 4- and 8-PAM.

STAGE-3

In this stage, we classify QAM constellation set or PSK constellation set into two

groups depending on the sub-classification at Stage 2. In QAM case, we use the

feature M̃42 and in PSK case, we use |M̃40,dp|. We use distance metric choosing the

thresholds as given in Fig. 3.1 and classify into two groups.

STAGE-4

In this stage, we perform inter classification between 16 and 32-QAM or between

O-QPSK and 4-PSK depending on the classification of the incoming signal at Stage

3. We use the feature |M̃40,dp| for classification in QAM and T̃ for classification in

the PSK case. We use distance metric choosing the thresholds as given in Fig. 3.1.

Chapter 4

Results

4.1 Simulation Results

We conducted simulations with data size of 1000 symbols with 2 samples per symbol,

and chose ε = 0.01. For each constellation, we carried out 105 trials, with different

realization of symbol sequence and noise sequence in each trial. From the number of

correct classifications obtained over 105 trials, the percentage of correct classification

is computed. Table 4.1 gives this value for all the constellations for various SNR

values. Note here that since the features used are insensitive to the carrier frequency

offset, the results will be same for all values of ε. The quantities to be stored in

the library are the thresholds given in Fig 3.1. The results show that our approach

performs very well for the SNR values ≥ 10dB. Even for lower SNR values of 6 to 10

dB, the performance is very good for all except the 32-QAM.

Fig. 4.1 gives the average percentage of correct classification, computed by aver-

aging over all the constellations. In [4], the average percentage of correct classification

for data size of 1000 symbols is given as 76 percent (approximately) at SNR of 10

dB. The corresponding value from our algorithm is about 95 percent. Of course, this

may not be a fair comparison since [4] considered 10 modulations while we considered

9 modulations of which only 8 are common to both, and the pulse shapes used in

both cases are different. Nevertheless, we wanted to point out this to convey that the

15

CHAPTER 4. RESULTS 16

Table 4.1: Performance of the proposed algorithm as percentage of correct classification(1000 symbols with two samples per symbol and ε=0.01 are used in 105 trials)

SNR 2-PAM 4-PAM 8-PAM 4-PSK O-QPSK 8-PSK 16-QAM 32-QAM 64-QAM20 100 96.341 96.439 100 100 100 95.709 98.982 96.8618 100 96.248 96.442 100 100 100 95.428 98.78 96.62516 100 96.282 96.297 100 100 100 95.068 98.462 96.35214 100 96.019 96.153 100 100 100 93.936 97.411 95.70412 100 95.895 95.487 100 100 100 92.214 94.552 94.68310 100 95.133 94.729 100 100 100 88.298 85.361 92.6138 100 93.833 92.958 100 100 100 80.733 61.578 88.5926 100 90.81 89.257 99.789 99.835 97.832 72.653 28.601 82.0214 100 85.335 82.712 93.481 93.369 54.787 69.138 7.967 73.1442 100 76.723 73.419 91.837 91.943 11.229 64.289 1.295 64.3060 99.565 66.675 64.199 88.001 88.908 1.246 50.109 0.123 57.363

Figure 4.1: Average percentage of correct classification

features used here have smaller variance and this resulted in improved performance.

4.2 10-Class Problem

We now consider a 10-Class problem consisting of 2-, 4- and 8-PAM, 4-PSK, O-

QPSK (Offset-QPSK), 8-PSK and 16-PSK, 16-, 32- and 64-QAM. All the considered


constellations are same as that of 9-class problem except 16-PSK. The asymptotic

values of various features used in the 9-class problem for 16-PSK are same as that of 8-

PSK as shown in Table 4.2. We, therefore, used eighth-order moment of differentially

processed signal as the feature for interclassification between 8-PSK and 16-PSK.

This feature is defined as

M80,Y2(n) = E[Y 82 (n)]

Using (2.8) and simplifying, we get

M80,X2(n) =e−j32πεM80,Y2(n)

(M21,y(n) − σ2w)8

(4.1)

The phase offset is removed by taking the absolute value. The estimated nor-

malised eighth order moment is given by

|M̃80,dp| =|M̂80,Y2(n)|

(M̂21,y(n) − σ2w)8

(4.2)

The asymptotic value of |M̃80,dp| is |M80,X2(n)| and Table 4.2 gives this value for

8-PSK and 16-PSK.

Table 4.2: Moments of x(n) and X2(n) for 8- & 16-PSK ( x(n) is of unit variance )Constellations M42,x(n) |M20,X2(n)| |M40,X2(n)| |M80,X2(n)|

8-PSK 1 0 0 116-PSK 1 0 0 0

We follow the same classification procedure as in 9-Class problem for classify-

ing constellations and use |M̃80,dp| for inter-classification of 8-PSK and 16-PSK. The

asymptotic value of |M̃80,dp| is 1 for 8-PSK and 0 for 16-PSK. Let H0 and H1 represent

the hypotheses that the incoming signal is from 8-PSK and 16-PSK, respectively. We

apply the distance metric choosing the threshold as 0.5. From the received data, we

evaluate |M̃80,dp|. If this is greater than or equal to 0.5 we choose hypothesis H0, oth-

erwise H1. The complete algorithm is given in Fig 4.2 and the corresponding results


Figure 4.2: Algorithm for hierarchical modulation classification using distance metricfor 10-Class problem (modulations considered are PAM, PSK and QAM)

in Table 4.3.

Table 4.3: Performance of the proposed algorithm as percentage of correct classificationfor 10-class problem (1000 symbols with two samples per symbol and ε=0.01 are used in105 trials)SNR 2-PAM 4-PAM 8-PAM 4-PSK O-QPSK 8-PSK 16-PSK 16-QAM 32-QAM 64-QAM

20 100 96.341 96.439 100 100 100 100 95.709 98.982 96.8618 100 96.248 96.442 100 100 100 100 95.428 98.78 96.62516 100 96.282 96.297 100 100 100 100 95.068 98.462 96.35214 100 96.019 96.153 100 100 99.992 99.891 93.936 97.411 95.70412 100 95.895 95.487 100 100 96.811 77.428 92.214 94.552 94.68310 100 95.133 94.729 100 100 90.951 17.765 88.298 85.361 92.6138 100 93.833 92.958 100 100 98.649 1.488 80.733 61.578 88.5926 100 90.81 89.257 99.789 99.835 97.772 0.057 72.653 28.601 82.0214 100 85.335 82.712 93.481 93.369 54.785 0.001 69.138 7.967 73.1442 100 76.723 73.419 91.837 91.943 11.229 0 64.289 1.295 64.3060 99.565 66.675 64.199 88.001 88.908 1.246 0 50.109 0.123 57.363


We observe that the classification of 8-PSK and 16-PSK is good for SNR values

greater than 10dB. Since the eighth order feature has higher variance compared to

that of fourth-order for a given data size, SNR required will be higher for obtaining

above average probability of correct classification.

Chapter 5

Conclusion

We addressed the problem of blind modulation classification with finite data in the

presence of carrier frequency offset. A hierarchical algorithm for classification is

proposed. Unlike in the recently reported works which focus more on cumulants, we

used moments as features while providing some motivation for their use. Simulation

results are given to illustrate the classification performance with our method.

5.1 Future Work

The problem of blind modulation classification becomes difficult when we simulta-

neously have several imperfections such as timing offset, multipath fading channels

apart from carrier frequency offset. So, development of methods for modulation clas-

sification in the combined presence of these imperfections is suggested as the future

work.

20

Appendix A

Evaluating Feature Used for 4-PSK & O-QPSK Classification

Let the four symbols of 4-PSK/ O-QPSK constellation be given by A, B, C and D.

Calculation of TX1(n) for 4-PSK

TX1(n) = E[|X1(n)−X1(n− 1)|2]

where

X1(n) = x∗(n− 1)x(n)

Case 1

Let n − 2, n − 1, n denote instances of time. Let us consider that symbol A has

come in the (n − 2)th instance from among the four symbols. Then the symbol at

(n − 1)th instance would also be A (because we are considering two samples per

symbol). Symbol at (n)th instance can be A, B, C or D. These transitions along with

the respective probabilities are given in Table A.1 .

Table A.1: Symbol transitions along with respective probabilities for 4-PSKn− 2 n− 1 n Probability

A A A [1/4]A A B [1/4]A A C [1/4]A A D [1/4]

21

APPENDIX A. EVALUATING FEATURE USED FOR 4-PSK & O-QPSK CLASSIFICATION22

Table A.2: X1(n) and X1(n− 1) along with respective probabilities for 4-PSKX1(n) X1(n− 1) Probability|A|2 |A|2 [1/4]A∗B |A|2 [1/4]A∗C |A|2 [1/4]A∗D |A|2 [1/4]

Now X1(n) is obtained by multiplying second(with conjugate) and third columns

and X1(n−1) is obtained by multiplying first(with conjugate) and second columns in

Table A.1. The resulting sequences are given in Table A.2. Now |X1(n)−X1(n− 1)|is obtained by subtracting first and second columns in Table A.2.

Table A.3: Absolute difference along with respective probabilities for 4-PSK|X1(n)−X1(n− 1)| Probability

0 [1/4]√2 [1/4]2 [1/4]√2 [1/4]

Case 2

Let n− 2, n− 1, n denote instances of time. Let us consider that symbol A has come

in the (n− 2)th instance from among the four symbols. Then the symbol at (n− 1)th

instance can be any one of A, B, C or D. Symbol at (n)th instance would be same as

that of (n − 1)th instance. These transitions along with the respective probabilities

are given in Table A.4 .

Table A.4: Symbol transitions along with respective probabilities for 4-PSKn− 2 n− 1 n Probability

A A A [1/4]A B B [1/4]A C C [1/4]A D D [1/4]


and X1(n−1) is obtained by multiplying first(with conjugate) and second columns in


Table A.5: X1(n) and X1(n− 1) along with respective probabilities for 4-PSKX1(n) X1(n− 1) Probability|A|2 |A|2 [1/4]|B|2 A∗B [1/4]|C|2 A∗C [1/4]|D|2 A∗D [1/4]

Table A.6: Absolute difference along with respective probabilities for 4-PSK|X1(n)−X1(n− 1)| Probability

0 [1/4]√2 [1/4]2 [1/4]√2 [1/4]

Table A.4. The resulting sequences are given in Table A.5. Now |X1(n)−X1(n− 1)|is obtained by subtracting first and second columns in Table A.5 and is provided in

Table A.6.

Case 1 occurs with probability (1/2) and Case 2 with probability (1/2). So TX1(n)

will be

1

2

[1

4× 0 +

1

4× 2 +

1

4× 4 +

1

4× 2

]+

1

2

[1

4× 0 +

1

4× 2 +

1

4× 4 +

1

4× 2

]

which is 2.

Due to symmetry, symbols B, C and D at (n− 2)th instance would also give value

of TX1(n)=2. All the four symbols occur with probability of [1/4]. So net value of

TX1(n)=2.

Calculation of TX1(n) for O-QPSK

TX1(n) = E[|X1(n)−X1(n− 1)|2]

where

X1(n) = x∗(n− 1)x(n)


Let n − 2, n − 1, n denote instances of time. Let us consider that symbol A

has come in the (n − 2)th instance from among the four symbols. Then the symbol

at (n)th instance can be any one of A, B, C or D each with probability [1/4]. The

transition from (n−2)th instance to (n)th instance occurs via (n−1)th instance. These

transitions along with the respective probabilities are given in Table A.7 .

Table A.7: Symbol transitions along with respective probabilities for O-QPSKn− 2 n− 1 n Probability

A A A [1]×[1/4]A B B [1/2]×[1/4]A A B [1/2]×[1/4]A B C [1/2]×[1/4]A D C [1/2]×[1/4]A D D [1/2]×[1/4]A A D [1/2]×[1/4]

Table A.8: X1(n) and X1(n− 1) along with respective probabilities for O-QPSKX1(n) X1(n− 1) Probability|A|2 |A|2 [1/4]|B|2 A∗B [1/8]A∗B |A|2 [1/8]B∗C A∗B [1/8]D∗C A∗D [1/8]|D|2 A∗D [1/8]A∗D |A|2 [1/8]

Table A.9: Absolute difference along with respective probabilities for O-QPSK|X1(n)−X1(n− 1)| Probability

0 [1/4]√2 [1/8]√2 [1/8]0 [1/8]0 [1/8]√2 [1/8]√2 [1/8]



and X1(n− 1) is obtained by multiplying first (with conjugate) and second columns

in Table A.7. The resulting sequences are given in Table A.8.

Now |X1(n) −X1(n − 1)| is obtained by subtracting first and second columns in

Table A.8 and is given in Table A.9.

Using |X1(n)−X1(n− 1)|, the absolute difference and probabilities,

TX1(n) = E[|X1(n)−X1(n− 1)|2] =1

2× (0)2 +

4

8× (√

2)2 = 1

Due to symmetry, symbols B, C and D at (n− 2)th instance would also give value

of TX1(n)=1. All the four symbols occur with probability of [1/4]. So net value of

TX1(n)=1.

Appendix B

Derivation of Moments of Differentially Processed Signal

We derive the moments of differentially processed received signal used in the classifi-

cation procedure. The received signal y(n) from (1.1) is

y(n) =√

Pej(2πεn+θ)x(n) + w(n) (B.1)

Y2(n) = y∗(n − 2)y(n) denotes the differentially processed received signal and

similarly X2(n) = x∗(n − 2)x(n). Combining (1.1) with the above definition, we

obtain

Y2(n) = Pej4πεx∗(n− 2)x(n) + (√

Pe−j(2πε(n−2)+θ)x∗(n− 2)

w(n)) +√

Pej(2πεn+θ)x(n)w∗(n− 2) + w∗(n− 2)w(n)

which can be expressed as


Pe−j(2πε(n−2)+θ)x∗(n− 2)w(n)

+√

Pej(2πεn+θ)x(n)w∗(n− 2) + w∗(n− 2)w(n) (B.2)

We relate the moments of X2(n) to those of Y2(n). From the definition,

M20,Y2(n) = E[Y2(n)Y2(n)] = E[Y 22 (n)]

26

APPENDIX B. DERIVATION OF MOMENTS OF DIFFERENTIALLY PROCESSED SIGNAL27

Substituting (B.2) for Y2(n) we get

M20,Y2(n) = E[(Pej4πεX2(n) +√

Pe−j(2πε(n−2)+θ)x∗(n− 2)w(n)

+√

Pej(2πεn+θ)x(n)w∗(n− 2) + w∗(n− 2)w(n))2]

On expanding the above equation we get

M20,Y2(n) = E[P 2ej8πεX22 (n) + Pe−j(4πε(n−2)+2θ)(x∗(n− 2))2w2(n)

+Pej(4πεn+2θ)x2(n)(w∗(n−2))2+(w∗(n−2))2w2(n)+2(Pej4πεX2(n))(√

Pe−j(2πε(n−2)+θ)x∗(n−2)

w(n))+2(√

Pej(2πεn+θ)x(n)(w∗(n−2))2w(n))+2(Pej4πεX2(n))(√

Pej(2πεn+θ)x(n)w∗(n−2))

+2(Pej4πεX2(n))(w∗(n−2)w(n))+2(√

Pe−j(2πε(n−2)+θ)x∗(n−2)w(n))(√

Pej(2πεn+θ)x(n)

w∗(n− 2)) + 2(√

Pe−j(2πε(n−2)+θ)x∗(n− 2)w(n))(w∗(n− 2)w(n))]

Taking the expectation of each term individually

M20,Y2(n) = E[P 2ej8πεX22 (n)] + E[Pe−j(4πε(n−2)+2θ)(x∗(n− 2))2w2(n)]

+E[Pej(4πεn+2θ)x2(n)(w∗(n−2))2]+E[(w∗(n−2))2w2(n)]+2E[(P32 ej4πεX2(n))(e−j(2πε(n−2)+θ)

x∗(n− 2)w(n))] + 2E[(√

Pej(2πεn+θ)x(n)(w∗(n− 2))2w(n))] + 2E[(P32 ej4πεX2(n))

(ej(2πεn+θ)x(n)w∗(n−2))]+2E[(Pej4πεX2(n))(w∗(n−2)w(n))]+2E[(Pe−j(2πε(n−2)+θ)x∗(n−2)

w(n))(ej(2πεn+θ)x(n)w∗(n− 2))] + 2E[(√

Pe−j(2πε(n−2)+θ)x∗(n− 2)w∗(n− 2)w2(n)]

From the assumptions underlying the signal model: a) the noise and the signal

are uncorrelated and b) the noise is white, all the terms other than the first term are

zero. Hence we have

M20,Y2(n) = P 2ej8πεE[X22 (n)]

APPENDIX B. DERIVATION OF MOMENTS OF DIFFERENTIALLY PROCESSED SIGNAL28

From definition we have

M20,X2(n) = E[X22 (n)]

Therefore

M20,Y2(n) = P 2ej8πεM20,X2(n)

M20,X2(n) =e−j8πεM20,Y2(n)

P 2

Substituting (2.4) in above equation we have

M20,X2(n) =e−j8πεM20,Y2(n)

(M21,y(n) − σ2w)2

(B.3)

To counter the phase offset we work with |M20,X2(n)|.

|M20,X2(n)| =|M20,Y2(n)|

(M21,y(n) − σ2w)2

(B.4)

By following similar approach we can show that

|M40,X2(n)| =|M40,Y2(n)|

(M21,y(n) − σ2w)4

(B.5)

|M80,X2(n)| =|M80,Y2(n)|

(M21,y(n) − σ2w)8

(B.6)

where M40,Y2(n) involves evaluation of 35 terms and M80,Y2(n) involves evaluation

of 165 terms.

Related Publication

V. Chaithanya and V. U. Reddy, “Blind Modulation Classification in the Presence of

Carrier Frequency Offset ,” Proceedings of the 8th International Conference on Signal

Processing and Communications (IEEE-SPCOM 2010), Indian Institute of Science,

Bangalore, 18-21 July, 2010.

29

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