Statistical Methods in Modulation Classification

TAMPERE UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF INFORMATION TECHNOLOGY

Antti-Veikko Rosti

STATISTICAL METHODS IN MODULATION CLASSIFICATION MASTER OF SCIENCE THESIS

SUBJECT APPROVED BY DEPARTMENTAL COUNCIL ON December 9, 1998 Examiners: Prof. Visa Koivunen Lic. Tech. Jukka Mannerkoski

Preface

The research and study presented in this thesis was carried out in the Signal Processing Laboratory of Tampere University of Technology. I would like to thank Professor Visa Koivunen, my advisor and examiner, for his valuable guidance in statistical signal processing and making of this thesis. I would also like to thank my other examiner, Lic. Tech. Jukka Mannerkoski, for sharing his expertise in cyclostationary statistics. I wish to express my gratitude to my colleagues, Mr. Anssi Rämö and Mr. Teemu Saarelainen, for their help in making of this thesis and putting up with me in the same oce. I am also grateful to Mr. Anssi Huttunen and Mr. Saarelainen for their help in proof reading. I would like to thank the people in Signal Processing Laboratory for the pleasant working environment. Especially I would like to thank Audio Research Group, where I hope to enjoy working in the future. I wish to express my gratitude to my family and friends who have always been there for me.

Tampere, June 1, 1999

Antti-Veikko Rosti Opiskelijankatu 4 E 290 33720 Tampere Tel. 050-523 7764

ii

Contents Abstract

vi

Tiivistelmä

vii

List of Abbreviations

viii

List of Symbols

x

1 Introduction

1

2 Representation of Communication Signals

3

2.1

2.2

2.3

Signal Representation . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1.1

Analytic Signal . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1.2

Complex Envelope . . . . . . . . . . . . . . . . . . . . . . .

6

Analog Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1

Amplitude Modulation . . . . . . . . . . . . . . . . . . . . .

9

2.2.2

Double-Sideband Modulation . . . . . . . . . . . . . . . . . 10

2.2.3

Single-Sideband Modulation . . . . . . . . . . . . . . . . . . 12

2.2.4

Frequency Modulation . . . . . . . . . . . . . . . . . . . . . 14

2.2.5

Phase Modulation . . . . . . . . . . . . . . . . . . . . . . . . 16

Digital Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1

Amplitude Shift Keying . . . . . . . . . . . . . . . . . . . . 18

2.3.2

Phase Shift Keying . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.3

Quadrature Amplitude Modulation . . . . . . . . . . . . . . 21

2.3.4

Frequency Shift Keying . . . . . . . . . . . . . . . . . . . . . 23

iii

3 Statistical Tools 3.1

3.2

25

Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1

Stationary Processes . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2

Cyclostationary Processes . . . . . . . . . . . . . . . . . . . 28

Higher-Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.1

Higher-Order Moments . . . . . . . . . . . . . . . . . . . . . 31

3.2.2

Cumulants and Multi-Correlations

3.2.3

Higher-Order Spectra . . . . . . . . . . . . . . . . . . . . . . 36

4 Review of Modulation Classication 4.1

4.2

. . . . . . . . . . . . . . 32

38

Maximum Likelihood Approach . . . . . . . . . . . . . . . . . . . . 38 4.1.1

General Maximum Likelihood Methods . . . . . . . . . . . . 39

4.1.2

MPSK Classier Based on the Exact Phase Distribution . . 39

4.1.3

Classiers Based on the Likelihood Functions . . . . . . . . 40

4.1.4

Maximum Likelihood Classier for CPM . . . . . . . . . . . 42

Pattern Recognition Approach . . . . . . . . . . . . . . . . . . . . . 42 4.2.1

Envelope Based Methods . . . . . . . . . . . . . . . . . . . . 43

4.2.2

Higher-Order Methods . . . . . . . . . . . . . . . . . . . . . 44

4.2.3

Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Implemented Methods

47

5.1

Test Signal Generation for the Simulations . . . . . . . . . . . . . . 47

5.2

Implemented Methods . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.1

Ratio of Variance to Squared Mean . . . . . . . . . . . . . . 48

5.2.2

Deviations in Instantaneous Properties . . . . . . . . . . . . 50

5.2.3

Even Moments of MPSK Signals

5.2.4

Time-Average of Complex Envelope MFSK Process . . . . . 55

6 Simulation Results 6.1

6.2

. . . . . . . . . . . . . . . 52

57

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1.1

Ratio of Variance to Squared Mean . . . . . . . . . . . . . . 57

6.1.2

Deviations in Instantaneous Properties . . . . . . . . . . . . 57

6.1.3

Even Moments of MPSK Signals

6.1.4

Time-Average of Complex Envelope MFSK Process . . . . . 63

. . . . . . . . . . . . . . . 62

Discussion on Features . . . . . . . . . . . . . . . . . . . . . . . . . 64 iv

7 Conclusions

66

References

68

A First and Second-Order Statistics of Digital Modulated Signals

72

A.1 Carrier Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 A.2 Amplitude Shift Keying . . . . . . . . . . . . . . . . . . . . . . . . 73 A.3 Phase Shift Keying . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.4 Frequency Shift Keying . . . . . . . . . . . . . . . . . . . . . . . . . 77

v

TAMPERE UNIVERSITY OF TECHNOLOGY Department of Information Technology Signal Processing Laboratory

Rosti, Antti-Veikko: Statistical Methods in Modulation Classication Master of Science Thesis, 71 pages and Appendix, 8 pages. Examiners: Prof. Visa Koivunen and Lic. Tech. Jukka Mannerkoski June 1999 Keywords: modulation classication, cyclostationarity, higher-order statistics

The interest in modulation classication has recently emerged in the research of communication systems. This has been due to the advances in recongurable signal processing systems, especially in the study of software radio. Published methods can be divided into two groups: maximum likelihood and pattern recognition approaches. In the maximum likelihood approach, decision rules are often simple but the test statistics are complicated and assume prior knowledge about the signals. In the pattern recognition approach, decision rules are often complicated whereas the features are simple and fast to calculate. Communication signals contain a vast amount of uncertainty due to the unknown modulating signal, modulation type, and noise. Therefore the modulation classication problem has to be approached by using statistical methods. The features and the test statistics may be derived from the known statistical characteristics of the modulated signals. Either implicit or explicit use of higher-order statistics has been studied previously in many communication applications. The higher-order statistics are often more preferable because second-order statistics suppress the phase information of the signal. Nevertheless, the estimation of the higher-order statistics requires long sample sets and has a high computational complexity. To overcome these problems, second-order cyclostationary statistics have been studied and the results seem promising. In this thesis, the feature extraction problem of the modulation classication is discussed. Useful characteristics and representations of the communication signals are presented as well as the relevant knowledge of statistical signal processing. The previous methods are presented in a literature review of the modulation classication. The rst and second-order statistics including the cyclostationary statistics of digital modulated signals are studied, and a novel feature is proposed. Some previous methods and this novel feature are compared by investigating their discrimination performance in Matlab simulations.

vi

TAMPEREEN TEKNILLINEN KORKEAKOULU Tietotekniikan osasto Signaalinkäsittely

Rosti, Antti-Veikko: Statistical Methods in Modulation Classication Diplomityö, 71 s. ja 8 liites. Tarkastajat: Prof. Visa Koivunen ja TkL Jukka Mannerkoski Kesäkuu 1999 Avainsanat: modulaation luokittelu, syklostationaarisuus, korkeamman asteen tunnusluvut

Tietoliikennejärjestelmien tutkimuksessa kiinnostus modulaation luokittelua kohtaan on herännyt viimeaikoina. Tämä johtuu uudelleenkonguroitavien signaalinkäsittelyjärjestelmien kehityksestä ja erityisesti ohjelmistoradion tutkimuksesta. Tunnetut menetelmät voidaan jakaa kahteen ryhmään lähestymistavan perusteella: suurimman uskottavuuden estimointiin ja hahmontunnistukseen. Suurimman uskottavuuden estimaatti voidaan usein päättää helposti mutta testattavat tunnusluvut ovat monimutkaisia ja signaaleista on tunnettava eräitä ominaisuuksia etukäteen. Hahmontunnistuksessa päätössäännöt ovat yleensä monimutkaisia mutta luokitteluun käytettävät piirteet ovat yksinkertaisia ja voidaan laskea nopeasti. Tietoliikennesignaaleihin liittyy paljon epävarmuutta, koska moduloivaa signaalia, modulaatiotapaa ja kohinaa ei tunneta. Tästä syystä modulaation luokitteluongelmaa täytyy käsitellä tilastollisin menetelmin. Piirteet tai testattavat tunnusluvut voidaan johtaa moduloitujen signaalien tilastollisista ominaisuuksista. Useissa tietoliikennesovelluksissa on tutkittu korkeamman asteen tunnuslukujen käyttöä joko implisiittisesti tai eksplisiittisesti. Niiden käyttö on suotuisaa, koska toisen asteen tunnusluvut kadottavat signaalin sisältämän vaiheinformaation. Kuitenkin korkeamman asteen tunnuslukujen estimointi on monimutkaista ja vaatii suuren näytemäärän. Toisen asteen syklostationaarisia tunnuslukuja on tutkittu näiden ongelmien välttämiseksi ja tulokset ovat olleet lupaavia. Tässä diplomityössä käsitellään piirteenerotusongelmaa modulaation luokittelussa. Työssä esitetään tietoliikennesignaalien ominaispiirteitä ja signaalien esitystapoja sekä tarvittavat tiedot tilastollisesta signaalinkäsittelystä. Kirjallisuusselvityksessä esitellään aiemmin kehitettyjä menetelmiä modulaation luokitteluun. Työssä on tutkittu digitaalisesti moduloitujen signaalien ensimmäisen ja toisen asteen tunnuslukuja sekä syklostationaarisia tunnuslukuja. Lisäksi esitellään uusi digitaalisesti moduloitujen signaalien luokitteluun soveltuva piirre. Tämän piirteen ja eräiden kirjallisuudesta löytyvien piirteiden erottelukykyä tutkitaan Matlab-simulaatioilla.

vii

List of Abbreviations ALF ALLR ALRT AM ASK AWGN BPSK CAF CNR CPFSK CPM CS CW cdf DFT DSB FM FSK GLRT GMLC GMSK GSM HOS I ISI i.i.d. LF LLF LLN LS LSB M MAP MASK MCPFSK MFSK ML MPSK MSK

Average likelihood function Average log-likelihood ratio Average likelihood ratio test Amplitude modulation Amplitude shift keying Additive white Gaussian noise Binary phase shift keying Cyclic autocorrelation function Carrier-to-noise ratio Continuous phase frequency shift keying Continuous phase modulation Cyclostationary Carrier wave Cumulative distribution function Discrete Fourier transform Double-sideband amplitude modulation Frequency modulation Frequency shift keying General likelihood ratio test General maximum likelihood classier Gaussian minimum shift keying Global system for mobiles Higher-order spectra In-phase component Inter-symbol interference Independent and identically distributed Likelihood function Log-likelihood function Law of large numbers Least squares Lower-sideband amplitude modulation Number of symbols in digital modulation Maximum a posteriori probability M-ary amplitude shift keying M-ary continuous phase frequency shift keying M-ary frequency shift keying Maximum likelihood M-ary phase shift keying Minimum shift keying viii

MQAM OQPSK PAM PBC PM PMF PSD PSK PSP pdf Q QAM QPSK qLLRC SCD SLC SNR SOSE SSB USB WSS

M-ary quadrature amplitude modulation Oset quadrature phase shift keying Pulse amplitude modulation Phase-based classier Phase modulation Probability mass function Power spectral density Phase shift keying Per-survivor processing Probability density function Quadrature component Quadrature amplitude modulation Quadrature phase shift keying Quasi log-likelihood ratio classier Spectral correlation density Square-law classier Signal-to-noise ratio Sum of squared envelope Single-sideband amplitude modulation Upper-sideband amplitude modulation Wide-sense stationary

ix

List of Symbols Modulated Signals

Ac A[m] a[k] acn [k] a(t) C(ω) c(t) Fs fc f∆ fN [k] f (t) fˆ[k] g(t) h(t) j M Ns R(ω) r[k] r(t) r˜(t) s[m] s(t) T u(t) X(ω) x(t) x(t) ˙ v(t) w(t) Z(ω) z(t) φ∆ φˆN L [k] φuw [k] φ[m]

Amplitude of carrier Discrete symbol amplitude sequence Discrete instantaneous amplitude sequence Normalized centered amplitude sequence Instantaneous amplitude Spectrum of complex envelope Complex envelope signal Sampling frequency Carrier frequency Frequency deviation Normalized centered instantaneous frequency sequence Instantaneous frequency Estimated instantaneous frequency sequence Signal pulse in digital modulation Impulse response of a linear lter Imaginary unit Number of symbols in digital modulation Number of samples in a segment Spectrum of r(t) Discrete received signal sequence Received signal Hilbert transform of r(t) Discrete symbol sequence Continuous symbol function Length of symbol interval Unit step function Fourier transform of x(t) Modulating signal Time derivative of x(t) Contribution of additive noise to phase component Additive noise Spectrum of z(t) Analytic representation of r(t) Phase deviation Estimated nonlinear phase component sequence Unwrapped phase sequence Phase state sequence

x

φ(t) δ(t) µ ωc ω(t) ω∆

Instantaneous phase Dirac's delta function Modulation index Angular frequency of carrier Angular frequency Angular frequency deviation

Random Processes

C(t) CnX (ω1 , . . . , ωn−1 ) cX n cX n (τ1 , . . . , τn−1 ) cum(·) EX [·] FX (·) fX (·) Hi L(x|Hi ) MnX (ω1 , . . . , ωn−1 ) mX n P (·) PnX (ω1 , . . . , ωn−1 ) p(x|Hi ) RX (t1 , t2 ) RX (τ ) α RX (τ ) S[m] S(ω) T X X[k] X(t) x x(t) α γnX µX (t) σX 2 σX 2 σX (t) Φ[m] ΦX (ω) ΨX (ω)

Random complex envelope process nth-order polyspectrum of X(t) nth-order cumulant of X nth-order cumulant function of X(t) Cumulant operator Expectation operator with respect to X cdf of X pdf of X Hypothesis number i Likelihood function on Hi nth-order moment spectrum of X nth-order moment of X Probability operator nth-order coherency index of X Conditional pdf Autocorrelation function of X(t) Autocorrelation function of WSS process X(t) CAF of X(t) Random symbol sequence Power spectral density Set of real numbers Random variable Discrete-time random sequence Continuous-time random process Observation Sample function Cyclic frequency nth-order cumulant function of X(t) at origin Mean function of X(t) Standard deviation of X Variance of X Variance function of X(t) Random phase state sequence Characteristic function of X(t) Cumulative function of X(t)

xi

Features

mn P R γmax µC µa42 µf42 σa σaa σaf σap σdp

nth-order moment of instantaneous phase Power spectrum symmetry measure Ratio of variance to squared mean Spectral power density maximum Time-average of complex envelope Kurtosis of instantaneous amplitude Kurtosis of instantaneous frequency Standard deviation of instantaneous amplitude Standard deviation of absolute amplitude Standard deviation of instantaneous frequency Standard deviation of absolute phase Standard deviation of direct phase

xii

Chapter 1 Introduction The interest in modulation classication has been growing since the late eighties up to date. It has several possible roles in both civilian and military applications such as signal conrmation, interference identication, monitoring, spectrum management, and surveillance [42]. At the moment, the most attractive application area is software radio and other recongurable communication systems. Modulation classication is an intermediate step between signal detection and demodulation. In addition to modulation type, some other parameters should be estimated before successful demodulation. Modulation classiers, like general pattern recognition systems [38], consist of measurement, feature extraction, and decision parts. The measurement is obtained by a front-end which will receive the signal of interest and carry out some preprocessing such as ltering, down-conversion, equalization, and sampling. The feature extraction part reduces the dimensionality of the measurement by extracting the distinctive features which should be simple and fast to calculate. There are several ways to make the decision based on the obtained features such as decision functions, distance functions, and neural networks. The received signal to be classied according to its modulation type contains much uncertainty which should be encountered by statistical tools. Therefore the known methods are based on dierent statistics obtained from the received signal. These statistics can be derived from continuous-time signals and they hold for sampled discrete signals which may be processed by some digital device. Some known methods are based on the higher-order statistics of the received signal but they are often very complicated and dicult to obtain [28]. The methods exploiting the cyclostationarity of digital modulated signals have not yet attained much attention although cyclostationary statistics have shown many desirable properties in other elds of communication systems [13]. There are also many ad-hoc methods which can be eventually shown to contain implicit statistical information about the signal. In this study, we have concentrated on the problem of the feature extraction in the modulation classiers. The features are selected for the simulation part according to their applicability for the modulation types used in radio communication.

CHAPTER 1. INTRODUCTION

2

Therefore the modulation types under concern have a constant envelope and in the case of digital modulation, small symbol sets. The most of the known methods are reviewed and a new attractive feature is introduced as a result of studying the rst and second-order statistics of digital modulated signals. This thesis consists of seven chapters and one appendix. Useful characteristics and representations of the communication signals are introduced in Chapter 2. The minimal representation is presented and it is used over the remaining chapters. Each modulation type is reviewed and their key properties are derived to show their importance in the modulation classication. Chapter 3 presents the relevant knowledge of the statistics which is necessary to describe the uncertainty in the received communication signal. The interpretation of random processes is given including stationary and cyclostationary processes. The second part of the chapter is dedicated to higher-order statistics including higher-order moments, cumulants, and spectra. Chapter 4 reviews the previous methods found in literature and publications. The reviewed methods have been divided into two groups according to their approach [5]. First the maximum likelihood and then the pattern recognition methods are presented. The latter is found more promising for our objectives. The most attractive features are discussed in more detail and implemented in Chapter 5 with the new proposed feature. Due to diverse application areas, we have not concentrated on any particular real-world signals. Therefore the generation of articial signals is presented in the beginning of the chapter. The results of the simulations are gathered in Chapter 6. They are given in gures which show the discrimination eciency of each feature. In the end of the chapter, the results are discussed and some conclusions are made from the point of view. Conclusions and future work on the subject of this thesis are discussed in Chapter 7 and the derivations of the rst and second-order statistics of digital modulated signals including cyclostationary statistics are given in Appendix A. The deduction of the proposed feature is given in this appendix as well.

Chapter 2 Representation of Communication Signals In communication systems [6], transmitted signals have to be carried over a channel. Communication channel is a non-ideal physical environment which has a nite band-width; i.e. it is a band-pass system. The band-width is also limited by adjacent channels separated by their frequency content. The band-pass nature of the communication channels leads to restrictions in the band-width of the transmitted signal. Depending on the characteristics of the channel, eective communication requires a high-frequency sinusoidal carrier. The amplitude, phase, or frequency of this carrier is altered proportionally to the transmitted information signal. This operation is called modulation. Modulation types can be divided into two different groups depending on the transmitted signal. If the transmitted signal is continuous, it is called analog modulation. If the transmitted signal consists of a nite alphabet of discrete symbols, it is called digital modulation. In this chapter, we present the key properties of dierent modulation types. In the rst section, convenient representations of the communication signals are discussed.

2.1 Signal Representation The nature of the modulated signals leads to high sampling rates and an excessive amount of memory is required when the received signal is stored. Fortunately there are ways to lower the sampling rate and reduce the amount of memory needed. This can be achieved by using signal representations dierent from the directly sampled form. These representations have theoretical and also practical value because the phase information of the signal can be extracted by means of these methods. If the signal denoted by r(t) is real, then its Fourier transform R(ω) has the hermitian symmetry [6] as follows

CHAPTER 2. REPRESENTATION OF COMMUNICATION SIGNALS

4

R(−ω) = R∗ (ω), |R(−ω)| = |R(ω)|,

arg R(−ω) = − arg R(ω),

(2.1)

where ω denotes the angular frequency ω = 2πf . It follows from Equation (2.1) that the spectrum of a real signal contains redundant information. The spectrum of a real band-pass signal r(t) is depicted in Figure 2.1. |R(ω)|

ω

Figure 2.1: Spectrum of real band-pass signal. Dierent signal representations are discussed in almost any communication systems and signal analysis book [3, 6, 23, 31, 33]. Next subsections are based on these references and will present how to obtain dierent instantaneous properties of communication signals and how to extract some useful features out of them.

2.1.1 Analytic Signal The spectral redundancy of the received real band-pass signal can be reduced by using analytic representation also called pre-envelope. An analytic signal can be obtained by using a Hilbert transformer or a quadrature lter. The transformation though does not lower the required amount of memory. Fortunately the sampling frequency can be reduced to exactly the band-width of the received signal by downconverting the analytic signal. This representation is called complex envelope and it is discussed in the next subsection. First we want to reduce the spectral redundancies in a real band-pass signal r(t) given above. The Fourier transform of the new signal z(t) is

Z(ω) = 2R(ω)u(ω) = R(ω)[1 + sgn(ω)],

(2.2)

where u(ω) and sgn(ω) are dened as

 1 ,ω > 0 u(ω) = 12 , ω = 0 ,  0 ,ω < 0

(2.3)


5

and

 1 ,ω > 0 sgn(ω) = 2u(ω) − 1 = 0 , ω = 0 .  −1 , ω < 0

(2.4)

| Z(ω)|

ω

Figure 2.2: Spectrum of analytic band-pass signal. The spectrum of z(t) is depicted in Figure 2.2. In Equation (2.2) the spectrum Z(ω) comprises of the Fourier transform of the band-pass signal r(t) and the Fourier transform of its Hilbert transform which can be expressed as

1 1 R(ω)sgn(ω) ↔ jr(t) ∗ =j πt π

Z

∞ −∞

r(τ ) dτ = j r˜(t), t−τ

(2.5)

where the asterisk denotes convolution; i.e. r˜(t) is obtained by applying the original 1 band-pass signal r(t) to a quadrature lter h(t) = πt . Two most important Hilbert transform pairs are sin x ↔ − cos x and cos x ↔ sin x. Equations (2.2) and (2.5) lead to the analytic signal of the band-pass signal, z(t) = r(t) + j r˜(t). The digital processing of the analytic signal requires half the sampling rate required for the real signal because the analytic signal has information only in the right half of the spectrum. Yet it requires the same amount of memory because the analytic signal is complex. Spectra of sampled r(t) and z(t) are illustrated in Figure 2.3. Fs denotes the sampling frequency and the solid line represents region under half of the sampling frequency. | Z(ω)|

|R(ω)|

Fs

Fs ω

Fs

Figure 2.3: Spectra of sampled r(t) and z(t).

Fs

ω


6

2.1.2 Complex Envelope The sampling frequency can be decreased exactly to the band-width of the bandpass signal by using the complex envelope representation. The complex envelope c(t) is obtained from the analytic signal z(t) as follows

c(t) = z(t)e−jωc t = m(t) + jn(t),

(2.6)

where

m(t) = r(t) cos(ωc t) + r˜(t) sin(ωc t), n(t) = r˜(t) cos(ωc t) − r(t) sin(ωc t).

(2.7)

In Equation (2.6), it can be seen that the complex envelope is the frequency shifted version of the analytic signal z(t) as illustrated in Figure 2.4. | C(ω)|

Fs

Fs

ω

Figure 2.4: Spectrum of complex envelope of sampled real signal. The real and imaginary parts of c(t) are called the in-phase (I) and quadrature (Q) component, respectively. The instantaneous amplitude a(t), instantaneous phase φ(t), and instantaneous frequency f (t) can be easily obtained from the analytic and complex envelope representations. The instantaneous amplitude can be expressed as

a(t) = |z(t)| = = |c(t)| =

p

r2 (t) + r˜2 (t)

p m2 (t) + n2 (t).

(2.8)

The instantaneous amplitude can be similarly extracted from the sampled signal r[k], where k is the time index. The normalized centered instantaneous amplitude sequence acn [k] can be obtained from the instantaneous amplitude sequence a[k] as follows

acn [k] =

a[k] − 1, ma

(2.9)


7

where Ns 1 X ma = a[k], Ns k=1

(2.10)

and Ns is the number of samples in a segment. Normalization by ma is used to compensate the channel gain. The instantaneous phase of the signal can be expressed as

½ φ(t) =

arg z(t) . arg c(t) + ωc t

(2.11)

The instantaneous phase of the modulated signal comprises of the linear component contributed by the carrier frequency and the nonlinear component contributed by the modulating signal. In complex envelope representation, the linear phase component is not present due to the down-conversion. Otherwise the linear component of the instantaneous phase must be removed in order to obtain the important features of the modulated signal. If the carrier frequency fc is accurately known, the nonlinear phase component of the sampled signal r[k] can be estimated as follows

2πfc k φˆN L [k] = φuw [k] − , fs

(2.12)

where φuw [k] is the unwrapped phase sequence. If the carrier frequency is unknown it can be obtained by linear trend removal using least squares (LS) estimation where [3] the sum of squares,

ε=

Ns X £

¤2 φuw [k] − C1 k − C2 ,

(2.13)

k=1

is minimized. C1 and C2 are the parameters of a linear model. The linear model can be represented as φuw = Hc+v, where v is assumed to be the noise contributed by the nonlinear component and the other parameters are

   H= 

0 1 .. .

 1 1  ..  , .

(Ns − 1) 1

 φuw

  = 

φuw (1) φuw (2) .. . φuw (Ns )

   , 

¸ C1 . c= C2 ·

(2.14)


8

The columns of the matrix H are independent and H has a full rank. The least squares estimate can be expressed in matrix form as [37]

c = [H T H]−1 H T φuw .

(2.15)

The LS estimate gives the maximum likelihood (ML) estimate if the nonlinear component is Gaussian, which is not always the case. The derivative of the instantaneous phase is the angular frequency ω(t). The instantaneous frequency of the modulated signal can be expressed as

f (t) =

1 1 dφ(t) ω(t) = , 2π 2π dt

(2.16)

where ω(t) = 2πf (t). The numerical derivative can be obtained from the unwrapped phase sequence as follows

φuw [k + 1] − φuw [k] , fˆ[k] = Fs 2π

(2.17)

where Fs is the sampling frequency. The normalized centered instantaneous frequency for a digital modulated signal can be expressed as

fN [k] =

fˆ[k] − mf , rs

(2.18)

where Ns 1 X mf = fˆ[k], Ns k=1

(2.19)

and rs is the symbol rate.

2.2 Analog Modulation Analog modulation types can be further divided into two groups [6]: linear and exponential modulation or angle modulation. The properties of these groups are given in Table 2.1, where W refers to the band-width and X(ω) refers to the spectrum of the modulating signal. Signal-to-noise ratio (SNR) can be increased in


9

Table 2.1: Properties of linear and exponential modulation Linear Exponential Methods AM, DSB, SSB, VSB FM, PM Envelope Depends on modulating signal Constant Spectrum Frequency shifted X(ω) Complex ratio to X(ω) Band-width ≤ 2W > 2W SNR Depends on transmitting power Depends on band-width the linear modulation only by increasing the transmitting power; in the exponential modulation a suitable compromise can be found between band-width and SNR. The exponential modulation is used more commonly due to these properties. In subsequent sections, we derive the analytic signals of the analog modulation types and give examples of these types using a frame of speech, depicted in Figure 2.5, as the modulating signal. The instantaneous properties in the following gures are obtained from the complex envelope, so the contribution of the carrier is not present. The frequency of the carrier is ωc , the amplitude of the carrier is Ac and the modulation index, where applies, is µ. The modulating signal x(t) is assumed to be normalized such that |x(t)| ≤ 1. 1

0.8

0.9

0.6

0.8

0.4

0.7 Normalized amplitude

Modulating signal

Spectrum of the modulating signal 1

0.2

0

−0.2

0.6

0.5

0.4

−0.4

0.3

−0.6

0.2

−0.8

0.1

−1

0

20

40

60

80

100 Time / ms

120

140

160

180

200

0 −8000

−6000

−4000

−2000

0 Frequency / Hz

2000

4000

6000

8000

Figure 2.5: Modulating speech signal and its spectrum.

2.2.1 Amplitude Modulation Amplitude modulation (AM) is the simplest modulation scheme. It is formed by varying the amplitude of the carrier wave according to the modulating signal. The analytic representation of the amplitude modulated signal can be expressed as

z(t) = Ac [1 + µx(t)]ejωc t ,

(2.20)

i.e. a sum of the carrier signal and the modulating signal scaled by Ac µ and shifted in frequency by ωc . By the hermitian symmetry of the modulating signal,


10

Spectrum of the AM signal 1

0.9

0.8

Normalized amplitude

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 4.92

4.94

4.96

4.98

5 Frequency / Hz

5.02

5.04

5.06

5.08 5

x 10

Figure 2.6: Spectrum of AM signal. the band-width of the modulated signal is B = 2W as seen in Figure 2.6. The envelope of the AM signal can be expressed as

(2.21)

a(t) = Ac [1 + µx(t)],

i.e. the modulating signal is biased to positive values. It can be easily seen that demodulation of the AM signal requires only the detection of the envelope. If the modulating index µ ≤ 1, no phase shifts will occur. The instantaneous phase can be expressed as

" φ(t) = tan−1

#

Ac [1 + µx(t)] sin(ωc t) = ωc t, Ac [1 + µx(t)] cos(ωc t)

(2.22)

where common terms Ac [1 + µx(t)] can be removed because they are always nonnegative. The linear term contributed by the carrier wave remains, which implies that the instantaneous frequency is simply

f (t) =

1 d(ωc t) = fc . 2π dt

(2.23)

The amplitude modulated signal and its associated instantaneous properties are illustrated in Figure 2.7.

2.2.2 Double-Sideband Modulation Double-sideband modulation (DSB) is a special case of amplitude modulation where the carrier is suppressed. The analytic DSB modulated signal can be expressed as


11

Amplitude Modulation


2.5

2.5

2

2

1.5

Instantaneous amplitude

Modulated signal

1

0.5

0

−0.5

1.5

1

−1

0.5

−1.5

−2

−2.5

0

20

40

60

80

100 Time / ms

120

140

160

180

0

200

0

20

40

60


80

100 Time / ms

120

140

160

180

200

140

160

180

200

Amplitude Modulation 50

3 40

30

Instantaneous frequency / kHz

Instantaneous phase / rad

2

1

0

−1

−2

20

10

0

−10

−20

−30

−40

−3 0

20

40

60

80

100 Time / ms

120

140

160

180

200

−50

0

20

40

60

80

100 Time / ms

120

Figure 2.7: AM signal and its instantaneous amplitude, phase, and frequency.

z(t) = Ac x(t)ejωc t ,

(2.24)

i.e. the modulating signal scaled by Ac and shifted in frequency by ωc . The bandwidth is still the same as in the amplitude modulation as seen in Figure 2.8. The envelope will be the absolute value of the modulating signal and can be expressed as Spectrum of the DSB signal 1

0.9

0.8


0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 4.92

4.94

4.96

4.98

5 Frequency / Hz

5.02

5.04

5.06

5.08 5

x 10

Figure 2.8: Spectrum of DSB signal.


a(t) = Ac |x(t)|.

12

(2.25)

Demodulation of the DSB modulated signal cannot be carried out by using an envelope detector anymore. Therefore a down-converter is needed. There will be also discontinuities in the instantaneous phase caused by the zero-crossings of the modulating signal. Therefore, the instantaneous phase is obtained by

" φ(t) = tan−1

# ½ Ac x(t) sin(ωc t) ωc t , x(t) ≥ 0 = π + ωc t , x(t) < 0 Ac x(t) cos(ωc t)

¡ ¢ = u − x(t) π + ωc t,

(2.26)

where u(t) is the unit step function. In other words, there will be a phase shift of π when the modulating signal crosses zero. Common terms Ac x(t) cannot be removed because the phase information would be lost. This may be deduced with trigonometric identities, sin(x) = − sin(−x) and cos(x) = cos(−x), as follows

−|x(t)| sin(ωc t) sin(−ωc t) = = − tan(−ωc t) = tan(π + ωc t). −|x(t)| cos(ωc t) − cos(−ωc t)

(2.27)

The instantaneous frequency is found by using the fact that the derivative of the unit step function is the Dirac's delta function

f (t) = −

¡ ¢ x(t) ˙ δ − x(t) + fc , 2

(2.28)

where x(t) ˙ denotes the time derivative of x(t). The instantaneous frequency though contains impulses towards the negative derivative of the modulating signal when zero-crossing occurs and is a constant otherwise. The DSB modulated signal and its associated instantaneous properties are illustrated in Figure 2.9.

2.2.3 Single-Sideband Modulation Single-sideband modulation (SSB) requires the same band-width as the modulating signal. Thus it occupies only half the band-width compared to the AM and DSB signals as seen in Figure 2.10. SSB modulated signal can be obtained through the same reasoning as the analytic signal in Section 2.1.1 and its analytic representation is given by


13

Double−Sideband Modulation


1.5

1.5

1


Modulated signal

0.5

0

1

0.5

−0.5

−1

−1.5

0

20

40

60

80

100 Time / ms

120

140

160

180

0

200

0

20

40

60


80

100 Time / ms

120

140

160

180

200

140

160

180

200

Double−Sideband Modulation 50

3 40

30



2

1

0

−1

−2

20

10

0

−10

−20

−30

−40

−3 0

20

40

60

80

100 Time / ms

120

140

160

180

200

−50

0

20

40

60

80

100 Time / ms

120

Figure 2.9: DSB signal and its instantaneous amplitude, phase, and frequency.

© z(t) = Ac x(t) cos(ωc t) ∓ x˜(t) sin(ωc t)

(2.29)

ª +j[x(t) sin(ωc t) ± x˜(t) cos(ωc t)] , where the upper sign is used for the upper-sideband modulation (USB) and lower sign is used for the lower-sideband modulation (LSB). By the trigonometric identity, sin2 (x) + cos2 (x) = 1, the envelope of the SSB signal is in both cases given by Spectrum of the SSB signal 1

0.9

0.8


0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 4.92

4.94

4.96

4.98

5 Frequency / Hz

5.02

5.04

5.06

5.08 5

x 10

Figure 2.10: Spectrum of LSB signal.


a(t) = Ac

14

p

(2.30)

x2 (t) + x˜2 (t).

The derivation of the instantaneous phase and frequency is more complicated than before because they depend heavily on the modulating signal. The instantaneous phase may be expressed as

( φ(t) = tan−1

) x(t) sin(ωc t) ± x˜(t) cos(ωc t) . x(t) cos(ωc t) ∓ x˜(t) sin(ωc t)

(2.31)

Single−Sideband Modulation


1.5

1.5

1


Modulated signal

0.5

0

1

0.5

−0.5

−1

−1.5

0

20

40

60

80

100 Time / ms

120

140

160

180

0

200

0

20

40

60


80

100 Time / ms

120

140

160

180

200

140

160

180

200

Single−Sideband Modulation 50

3 40

30



2

1

0

−1

−2

20

10

0

−10

−20

−30

−40

−3 0

20

40

60

80

100 Time / ms

120

140

160

180

200

−50

0

20

40

60

80

100 Time / ms

120

Figure 2.11: LSB signal and its instantaneous amplitude, phase, and frequency. The instantaneous frequency varies quite little with real continuous signals due to the smoothness of the instantaneous phase. The LSB modulated signal and its associated instantaneous properties are depicted in Figure 2.11.

2.2.4 Frequency Modulation The frequency modulation (FM) and phase modulation (PM) dier considerably from the previous linear modulation schemes. The major dierence is in the instantaneous amplitude which, in the linear modulation, varies depending on the


15

modulating signal and, in the exponential modulation, is a constant. In the exponential modulation, the angle of the carrier is altered with respect to the modulating signal. Thus exponential modulation is often called angle modulation. Spectrum of the FM signal 1

0.9

0.8


0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 4.5

4.6

4.7

4.8

4.9

5 5.1 Frequency / Hz

5.2

5.3

5.4

5.5 5

x 10

Figure 2.12: Spectrum of FM signal. Due to the exponential nature, the spectrum of the FM signal will be asymmetric and wide compared to the linear modulation schemes. The spectrum of a frequency modulated signal is depicted in Figure 2.12. The analytic FM signal may be expressed as

Z

h

t

z(t) = Ac exp {j ωc t + 2πf∆

i x(λ)dλ },

t0

(2.32)

where f∆ is the frequency deviation which represents the maximum shift of f (t) relative to the carrier frequency fc . The instantaneous amplitude is clearly a constant,

(2.33)

a(t) = Ac .

If we take t0 in Equation (2.32) such that φ(t0 ) = 0, the lower limit in the instantaneous phase can be dropped and the informal expression can be used as follows

Z φ(t) = ωc t + ω∆

t

x(λ)dλ,

(2.34)

where x(t) must be zero-mean so that the integrals do not diverge when t → ∞. By dierentiating the above expression with respect to t we get the instantaneous frequency as follows


16

Frequency Modulation


1.5

1.5

1


Modulated signal

0.5

0

1

0.5

−0.5

−1

−1.5

0

20

40

60

80

100 Time / ms

120

140

160

180

0

200

0

20

40

60


80

100 Time / ms

120

140

160

180

200

140

160

180

200

Frequency Modulation 50

3 40

30



2

1

0

−1

−2

20

10

0

−10

−20

−30

−40

−3 0

20

40

60

80

100 Time / ms

120

140

160

180

200

−50

0

20

40

60

80

100 Time / ms

120

Figure 2.13: FM signal and its instantaneous amplitude, phase, and frequency.

f (t) = fc + f∆ x(t).

(2.35)

Clearly the instantaneous frequency carries the message which can be extracted by removing the constant fc . The FM signal and its associated instantaneous properties are illustrated in Figure 2.13.

2.2.5 Phase Modulation In phase modulation (PM), the instantaneous phase is varied according to the modulating signal. The spectrum of a phase modulated signal is depicted in Figure 2.14. It is narrower compared to the FM signal due to the smoothness of the modulating signal. The analytic phase modulated signal may be expressed as

£ ¤ z(t) = Ac exp {j ωc t + φ∆ x(t) },

(2.36)

where φ∆ is the phase deviation which represents the maximum phase shift produced by x(t). Again, the instantaneous amplitude is a constant,


17

Spectrum of the PM signal 1

0.9

0.8


0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 4.92

4.94

4.96

4.98

5 Frequency / Hz

5.02

5.04

5.06

5.08 5

x 10

Figure 2.14: Spectrum of PM signal.

a(t) = Ac .

(2.37)

The instantaneous phase is obviously composed of the linear phase component and the modulating information signal as follows

φ(t) = ωc t + φ∆ x(t).

(2.38)

The upper bound for the phase deviation is φ∆ ≤ 180◦ which limits the instantaneous phase to the range ±π in order to avoid phase ambiguities. The instantaneous frequency can be expressed as

f (t) = fc +

1 φ∆ x(t), ˙ 2π

(2.39)

where x(t) ˙ denotes the time derivative of x(t) and causes only a small deviation due to the smooth modulating signal. The PM signal and its associated instantaneous properties are illustrated in Figure 2.15.

2.3 Digital Modulation A general analytic representation of digital modulated signals is given by [23]

jωc t

z(t) = Ac e

∞ X m=−∞

s[m]g(t − mT ),

(2.40)


18

Phase Modulation

Phase Modulation

1.5

1.5

1


Modulated signal

0.5

0

1

0.5

−0.5

−1

−1.5

0

20

40

60

80

100 Time / ms

120

140

160

180

0

200

0

20

40

60

Phase Modulation

80

100 Time / ms

120

140

160

180

200

140

160

180

200

Phase Modulation 50

3 40

30



2

1

0

−1

−2

20

10

0

−10

−20

−30

−40

−3 0

20

40

60

80

100 Time / ms

120

140

160

180

200

−50

0

20

40

60

80

100 Time / ms

120

Figure 2.15: PM signal and its instantaneous amplitude, phase, and frequency. where Ac is the amplitude and ωc is the frequency of the carrier. The discrete symbol sequence s[m] comprises of an alphabet distinctive for the modulation type. The elements of the alphabet are complex-valued points in the signal space. The waveform g(t) is a real-valued signal pulse whose shape inuences the spectrum of the modulated signal. The pulse shape reduces the large band-width caused by the discontinuities in the symbol sequence. In order to avoid inter-symbol interference (ISI), it is often required that g(0) = 1 and g(nt) = 0 for n = ±1, ±2, . . . . Such shapes are, e.g., sinc and raised cosine pulses.

2.3.1 Amplitude Shift Keying Amplitude shift keying (ASK) also known as pulse amplitude modulation (PAM) is the simplest digital modulation scheme. The alphabet consists of M = 2b points in the real line of the signal space where each point represents a sequence of b bits. Therefore the symbols are represented by dierent amplitude levels of the modulated signal. The analytic ASK modulated signal can be expressed using s[m] = (2n + 1 − M )d in Equation (2.40), where n ∈ [0, M − 1] is the nth symbol and 2d is the distance between adjacent signal amplitudes. The instantaneous amplitude of the ASK modulated signal can be expressed as

a(t) = Ac |s(t)|,

(2.41)


19

where

s(t) =

∞ X

(2.42)

s[m]g(t − mT ),

m=−∞

i.e., the absolute value of the symbol function s(t) with dierent amplitude levels scaled by Ac . The instantaneous phase is obtained similarly to DSB in analog modulation because the negative symbol values lead to phase shifts as follows

¡ ¢ φ(t) = u − s(t) π + ωc t,

(2.43)

where u(t) is the unit step function. The instantaneous frequency may be expressed as

f (t) = −

¡ ¢ s(t) ˙ δ − s(t) + fc , 2

(2.44) 2ASK/2PSK

2ASK/2PSK

1.5

1.5

1


Modulated signal

0.5

0

1

0.5

−0.5

−1

−1.5

0

200

400

600

800

1000 Time / us

1200

1400

1600

1800

0

2000

0

200

400

600

800

1000 Time / us

1200

1400

1600

1800

2000

1600

1800

2000

2ASK/2PSK 5

2ASK/2PSK 1500

4 1000



3

2

1

500

0

−500

0

−1000

−1

−2

0

200

400

600

800

1000 Time / us

1200

1400

1600

1800

2000

−1500

0

200

400

600

800

1000 Time / us

1200

1400

Figure 2.16: 2ASK/2PSK signal and its instantaneous amplitude, phase, and frequency. i.e. impulses in the instantaneous frequency occur at symbol transitions. The 2ASK modulated or two level PAM signal and its associated properties are illustrated in Figure 2.16.


20

2.3.2 Phase Shift Keying Phase shift keying (PSK) is obtained by dening a unique phase state of the carrier for every symbol as follows

s[m] = ejφ[m] ,

φ[m] ∈ {0,

(M − 1)2π 2π ,..., }, M M

(2.45)

where symbols do not have any eect in the instantaneous amplitude. The analytic PSK modulated signal may be expressed as

z(t) = Ac

∞ X

ej(ωc t+φ[m]) g(t − mT ).

(2.46)

m=−∞

Usual choices for M are 2, 4 and 8. 2PSK and 4PSK are commonly called binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK), respectively. Larger constellations are too dense and therefore not robust to noise. There is still some contribution of g(t) to the envelope but due to the properties of g(t) its sum will be approximately unity. The instantaneous amplitude is given by

a(t) = Ac

∞ X

g(t − mT ).

(2.47)

m=−∞

The instantaneous phase depends on the summation term m in Equation (2.46). Therefore the expression for the instantaneous phase is

φ(t) = ωc t +

h ³ ³ m−1 ´ m + 1 í φ[m] u t − T −u t− T , 2 2 m=−∞ ∞ X

(2.48)

where the unit step functions pick up the correct phase term in every time instant. The phase of the modulated signal consists of the phase states caused by the symbol sequence. The instantaneous frequency is obtained by ∞ ³ h ³ m + 1 í 1 X m−1 ´ T −δ t− T , f (t) = fc + φ[m] δ t − 2π m=−∞ 2 2

(2.49)

i.e. again impulses occur at symbol transitions. The 2PSK modulated signal and associated instantaneous properties are illustrated in Figure 2.16.


21

16QAM constellation 4

3

2

Imaginary

1

0

−1

−2

−3

−4 −4

−3

−2

−1

0 Real

1

2

3

4

Figure 2.17: Constellation of 16QAM.

2.3.3 Quadrature Amplitude Modulation Quadrature amplitude modulation (QAM) is a combination of ASK and PSK. The symbols are separated by both amplitude and phase dierences. Often the constellation is chosen to be square such as 16QAM constellation in Figure 2.17. QAM is mostly used in wired channels, e.g. cables, due to the larger number of the symbols and weaker tolerance for noise. Constellations are often chosen to be powers of two up to M = 256 or more. The symbols may be represented as complex numbers Re{s[m]}+j Im{s[m]}, where Re{·} denotes the real component and Im{·} denotes the imaginary component. For the extraction of the instantaneous properties, though, it is more convenient to express the symbols in polar coordinates as follows

s[m] = A[m]ejφ[m] ,

(2.50)

where

A[m] = |s[m]|,

and

φ[m] = arg s[m].

(2.51)

The analytic QAM signal is now given by

z(t) = Ac

∞ X

A[m]ej(ωc t+φ[m]) g(t − mT ),

(2.52)

m=−∞

where A[m] and g(t) are the only terms aecting on the envelope and A[m] ≥ 0, so the absolute value can be omitted. The instantaneous amplitude may be expressed as


a(t) = Ac

∞ X

22

(2.53)

A[m]g(t − mT ).

m=−∞

Again, because A[m] ≥ 0, it does not have any eect on the instantaneous phase. The instantaneous phase is obtained by

φ(t) = ωc t +

h ³ ³ m−1 ´ m + 1 í φ[m] u t − T −u t− T . 2 2 m=−∞ ∞ X

(2.54)

Due to discontinuities in the instantaneous phase the expression for the instantaneous frequency may be written as ∞ h ³ ³ 1 X m−1 ´ m + 1 í f (t) = fc + φ[m] δ t − T −δ t− T . 2π m=−∞ 2 2

16QAM 6

4

5

2

4


Modulated signal

16QAM 6

0

3

−2

2

−4

1

−6

0

200

400

600

800

1000 Time / us

(2.55)

1200

1400

1600

1800

0

2000

0

200

400

600

800

16QAM

1000 Time / us

1200

1400

1600

1800

2000

1800

2000

16QAM 100

3

50



2

1

0

−1

0

−50

−100

−2 −150

−3 0

200

400

600

800

1000 Time / us

1200

1400

1600

1800

2000

−200

0

200

400

600

800

1000 Time / us

1200

1400

1600

Figure 2.18: 16QAM signal and its instantaneous amplitude, phase, and frequency. The 16QAM signal and its associated instantaneous properties are illustrated in Figure 2.18. It can be observed that there are only three dierent amplitude levels. There should be twelve dierent phase states but there are not enough symbols in the example to show all of them.


23

2.3.4 Frequency Shift Keying Frequency shift keying (FSK) diers from the digital modulation schemes described so far due to the fact that it cannot be represented by Equation (2.40). The analysis of FSK will be a combination of methods given earlier and those used for the FM modulated signals in Section 2.2. The FSK modulated signal comprises of pulses having dierent frequencies depending on the symbol. Usual choices for the number of the dierent frequencies are 2, 4 and 8. The phase of the FSK signal can be continuous or discontinuous depending on the duration of the pulses. If there is an integer number of periods in every pulse, the phase of the signal will be continuous. The analytic FSK modulated signal may be expressed as follows

Z t h i z(t) = Ac exp {j ωc t + ω∆ s(τ )dτ },

(2.56)

where

s(t) =

∞ X

s[m]g(t − mT ),

(2.57)

m=−∞

and ω∆ is the frequency dierence of two adjacent pulses. The signal is called continuous-phase FSK (CPFSK) if the pulse shape g(t − mT ) is a square and f∆ = 1/T . Obviously the envelope of the FSK signal will be a constant (2.58)

a(t) = Ac .

The band-width of the FSK signals may be reduced by choosing f∆ = 1/(2T ) which is called minimum shift keying (MSK) and by choosing g(t) as low-pass lter with Gaussian shape we get Gaussian MSK (GMSK), which is used in global system for mobiles (GSM). The instantaneous phase is given by

Z φ(t) = ωc t + ω∆

t

s(τ )dτ,

(2.59)

where the value of the integral depends only on the pulse shape. The instantaneous frequency then becomes

f (t) = fc + f∆ s(t),

(2.60)

i.e. the instantaneous frequency varies with respect to the symbol values. The 2FSK modulated signal and its associated instantaneous properties are illustrated in Figure 2.19. There are also more complicated methods to implement the CPFSK signals exploiting memory and one or more modulation indices, hk , k ≥ 1. These form a general class of continuous-phase modulated (CPM) signals.


24

2FSK

2FSK

1.5

1.5

1


Modulated signal

0.5

0

1

0.5

−0.5

−1

−1.5

0

200

400

600

800

1000 Time / us

1200

1400

1600

1800

0

2000

0

200

400

600

800

2FSK

1000 Time / us

1200

1400

1600

1800

2000

1200

1400

1600

1800

2000

2FSK 14

3 12 2



10

1

0

−1

8

6

4

2 −2 0 −3 0

200

400

600

800

1000 Time / us

1200

1400

1600

1800

2000

−2

0

200

400

600

800

1000 Time / us

Figure 2.19: 2FSK signal and its instantaneous amplitude, phase, and frequency.

Chapter 3 Statistical Tools In communication systems, there is a vast amount of uncertainty present in the received signal due to unknown modulating signal and unknown modulation type. Hence, it is necessary to consider the received signal as a random process or a random sequence in the case of a sampled signal. In this chapter we present statistics that are useful for the modulation classication task. General denitions and well-known statistics of the random processes and random sequences may be found in [30, 37], for example.

3.1 Random Processes A random process X(t) is a rule for assigning a function X(t, ζ) to every ζ ∈ Ω, where Ω is the sample description space and ζ is an event. Thus, a random process is a family of time-functions depending on the parameter ζ or, equivalently, a function of t and ζ . The domain of ζ is the set of all experimental outcomes and the domain of t is a set T of real numbers. If T is the real axis, X(t) is a continuous-time process. If T is the set of integers, then X(t) is a discrete-time process. A discrete-time process is, thus, a sequence of random variables, also known as random sequence. Such a sequence will be denoted by X[k]. Three dierent sample functions xi (t), i = 1, 2, 3 are depicted on the left hand side of Figure 3.1. Each element of the sample description space ζi ∈ Ω maps to a continuous-time function; i.e. sample function. An example for the special case Ω = [0, ∞) is illustrated on the right hand side of the gure. For a specic t, X(t) is a random variable with distribution FX (x; t) = P (X(t) ≤ x). This function depends on t, and it equals the probability of the event X(t) ≤ x consisting of all outcomes ζi such that, at the specic time t, the samples X(t, ζi ) of the given process do not exceed the number x. The function FX (x, t) is called the rst-order distribution of the process X(t). The random process is said to be statistically specied by its nth-order cumulative distribution functions (cdf) for

CHAPTER 3. STATISTICAL TOOLS

26

X(t, ζ )

X(t,ζ1)

x1 (t)

X(t1, ζ1 )=x(t)

t

ζ1

ζ

ζ1

x2 (t) ζ2

t

A Sample Function

0

ζ3

x3 (t)

t1

t Ω

t

Figure 3.1: Random sample functions and special case Ω = [0, ∞). all n ≥ 1, and all times, t1 , . . . , tn , i.e. if we are given

FX (x1 , . . . , xn ; t1 , . . . , tn ) = P (X(t1 ) ≤ x1 , . . . , X(tn ) ≤ xn ).

(3.1)

The nth-order probability density functions (pdf) are given for dierentiable FX as

fX (x1 , . . . , xn ; t1 , . . . , tn ) = ∂ n FX (x1 , . . . , xn ; t1 , . . . , tn )/∂x1 . . . ∂xn .

(3.2)

Equivalently, we can statistically specify a random sequence by its nth-order distribution functions for all n ≥ 1, and all times, k, k + 1, . . . , k + n − 1, i.e. if we are given

FX (ak , ak+1 . . . , ak+n−1 ; k, k + 1, . . . , k + n − 1)

(3.3)

= P (X[k] ≤ ak , X[k + 1] ≤ ak+1 , . . . , X[k + n − 1] ≤ ak+n−1 ). The nth-order probability density functions are given for dierentiable FX as

fX (ak , ak+1 . . . , ak+n−1 ; k, k + 1, . . . , k + n − 1)

(3.4)

= ∂ n FX (ak , ak+1 . . . , ak+n−1 ; k, k + 1, . . . , k + n − 1)/∂ak ∂ak+1 . . . ∂ak+n−1 . The statistical specication of a random process, though, requires the knowledge of the function F (x1 , . . . , xn ; t1 , . . . , tn ) for every xi , ti and n. It may seem that this specication is some distance from a complete description of the entire random process compared to the direct specication in terms of X(t, ζ). However the word


27

statistical indicates that the former information can be obtained, at least conceptually, by estimating the nth-order cdf's for n = 1, 2, 3, . . . by using statistics. The rst moment or mean function of a random process is

Z

∞

µX (t) , E[X(t)] =

xf (x; t)dx,

−∞ < t < ∞.

−∞

(3.5)

Similarly the second moment or correlation function is dened as the expected value of the conjugate-product,

Z ∗

∞

Z

∞

RX (t1 , t2 ) , E[X(t1 )X (t2 )] = −∞

−∞

x1 x∗2 f (x1 , x2 ; t1 , t2 )dx1 dx2 , −∞ < t1 , t2 < ∞.

(3.6)

The value of RX (t1 , t2 ) on the diagonal t1 = t2 = t is the average power of X(t) because RX (t, t) = E[|X(t)|2 ]. The second central moment or covariance function is dened as the expected value of the conjugate-product of the centered process,

CX (t1 , t2 ) , E[(X(t1 ) − µX (t1 ))(X(t2 ) − µX (t2 ))∗ ] = RX (t1 , t2 ) − µX (t1 )µ∗X (t2 ).

(3.7)

2 Also variance function is dened as σX (t) , CX (t, t) = E[|X(t) − µX (t)|2 ].

3.1.1 Stationary Processes A random process X(t) is said to be stationary if it has the same nth-order distribution function as X(t + T ) for all T and for all positive n. Similarly, a random sequence X[k] is said to be stationary if for all positive n, the nth-order distributions do not depend on the shift parameter τ . This can be expressed by using density functions as

fX (x1 , . . . , xn ; t1 , . . . , tn ) = fX (x1 , . . . , xn ; t1 + T, . . . , tn + T ), fX (a0 , a1 . . . , an−1 ; τ, τ + 1, . . . , τ + n − 1) = fX (a0 , a1 . . . , an−1 ; 0, 1, . . . , n − 1). (3.8) By taking T = −t, it follows from above that the rst-order density is fX (x; t) = fX (x, 0) i.e. independent of t and the mean function is µX (t) = µX (0) , µX i.e. a constant. If we choose T = −t2 the second-order density is fX (x1 , x2 ; t1 , t2 ) =


28

fX (x1 , x2 ; t1 − t2 , 0) which depends only on the lag τ = t1 − t2 and the autocorrelation function can be expressed as RX (t1 , t2 ) = RX (t1 − t2 , 0) , RX (τ ). Similarly for the nth-order density T can be chosen in such a way that the density will depend only on the lags τi , i = 1, . . . , n − 1. A random process is called strict-sense stationary if all its density functions are invariant to a shift of origin. For most cases strict-sense stationary is too strong an assumption and it is often desirable to partially characterize a random process based on the knowledge of its rst two moments only, i.e. mean function and autocorrelation function. If the mean function is constant µX (t) , µX for all t and the autocorrelation function is shift-invariant RX (t1 − t2 , 0) , RX (τ ) for all t1 , t2 the random process is said to be wide-sense stationary (WSS). Thus the WSS property is considerably weaker than strict-sense stationarity. We note in particular that the average power of at least WSS process is also independent of t and equals E[|X(t)|2 ] = R(0). The power spectral density (PSD) of at least WSS process describes the frequency distribution of the signal power. The PSD, denoted by S(ω), of a process X(t) is the Fourier transform of its autocorrelation function R(τ ),

Z

∞

S(ω) =

R(τ )e−jωτ dτ,

−∞

(3.9)

which is the famous Wiener relation. Since R(−τ ) = R∗ (τ ), the PSD of a process X(t) is a real function of ω whether X(t) is real or complex. It means that the PSD does not contain any phase information which is a problem in system identication. Stationarity provides the possibility of learning the statistical properties under various ergodicity hypotheses [37]. Generally a statistical description of the random process is not available. Yet for many stationary processes, time-averages will tend to ensemble averages. Such processes are called ergodic.

3.1.2 Cyclostationary Processes Many man-made systems such as communication and control systems employ signal formats that have some form of periodic processing operation. Signals produced by samplers, scanners, multiplexors, and modulators are familiar examples. Often these signals are appropriately modeled by random processes that are cyclostationary (CS), i.e. processes with statistical parameters, such as mean and autocorrelation, that uctuate periodically with time. The processes that have cyclostationary statistics do not have to be periodic in time. The basic books of random processes often give little notice on the cyclostationary processes. Lately the exploitation of the cyclostationary statistics has grown due to some major benets e.g. in detection and channel identication. Extensive characterizations of cyclostationary processes are given by Gardner in [13] and Gardner and Franks in [11].


29

In communications, cyclostationarity often arises due to waveform repetition at the baud or symbol rate [11]. Therefore oversampling with respect to the symbol rate or several receivers are required. A continuous-time second-order random process {X(t); t ∈ (−∞, ∞)} is dened to be CS in the wide-sense, or of secondorder, with cycle period T if and only if its mean and autocorrelation exhibit the periodicity

µX (t) , E[X(t)] = µX (t + T ), RX (t1 , t2 ) , E[X(t1 )X ∗ (t2 )] = RX (t1 + T, t2 + T ).

(3.10)

It is later more convenient to work with the symmetric delay product with τ = t1 − t2 . Then the autocorrelation in Equation (3.10) can be rewritten as

τ τ τ τ τ τ RX (t + , t − ) = E[X(t + )X ∗ (t − )] = RX (t + + T, t − + T ). (3.11) 2 2 2 2 2 2 Since the autocorrelation function is periodic it can be expanded to its Fourier series representation,

X τ τ α RX (t + , t − ) = RX (τ )ej2παt , 2 2 α

(3.12)

where

α RX (τ )

1 , T

Z

∞

τ τ R(t + , t − )e−j2παt dt, 2 2 −∞

(3.13)

α and α is the cyclic frequency. RX is called cyclic autocorrelation function (CAF) which is a function of two variables τ and α. For a process that exhibits a single periodicity, the range of α is the set of integer multiples, i.e. harmonics of the 0 fundamental frequency. For α = 0 the Fourier series coecient RX (τ ) is equal to the time averaged probabilistic autocorrelation function RX (τ ), dened for a WSS process. Thus, a process X(t) is cyclostationary of second-order if and only α if RX (τ ) 6= 0 for any α 6= 0.

If the autocorrelation RX (t + τ2 , t − τ2 ) of a process X(t) is not periodic, it cannot α be expanded to a Fourier series. In that case RX can be dened as [12]

τ τ α (τ ) , hX(t + )X ∗ (t − )e−j2παt i, RX 2 2

(3.14)


30

where h·i is the time-averaging operation dened as

1 h·i , lim T →∞ T

Z

T /2

(·)dt. −T /2

(3.15)

α If RX (τ ) is nonzero for some α 6= 0, the process is called polycyclostationary (or multiply-cyclostationary or almost cyclostationary). α can be interpreted as a conventional crossThe cyclic autocorrelation function RX correlation function of two frequency shifted versions of X(t), namely U (t) , X(t)e−jπαt and V (t) , X(t)ejπαt . Now the CAF can be expressed as

τ τ α RU V (τ ) , hU (t + )V ∗ (t − )i = RX (τ ). 2 2

(3.16)

Similarly to the PSD of stationary process, we can dene the Fourier transform of the cyclic autocorrelation for a certain α as

Z α SX (ω)

∞

, −∞

α RX (τ )e−jωτ dτ,

(3.17)

which can be considered as a function of two variables ω and α, and it is called the spectral correlation density (SCD) function. There is a similar relation between the SCD at cyclic frequency α = 0 and averaged probabilistic PSD as between the CAF and the averaged probabilistic autocorrelation function.

3.2 Higher-Order Statistics Statistics can be used to characterize probability density functions and to estimate fX (x) from experimental measurements. The statistics can be determined analytically via the characteristic function of X . The characteristic function is given by [37, 30]

Z ΦX (ω) , E[e

jωX

∞

]=

fX (x)ejωx dx,

−∞

(3.18)

which except for a minus sign in the exponent, is the Fourier transform of fX (x). For the Gaussian random variable X with distribution N (µ, σ 2 ) the characteristic function can be shown to be

ΦX (ω) = exp(jµω −

σ2 2 ω ). 2

(3.19)


31

The inversion of the characteristic function can be obtained similarly to the inverse Fourier transform and can be expressed as

1 fX (x) , 2π

Z

∞

ΦX (ω)e−jωx dω.

(3.20)

−∞

Due to the inversion formula, knowing ΦX is equivalent to knowing fX (x) and vice versa. By the properties of the Fourier transform the characteristic functions gain one desirable property; i.e. the convolution theorem. Since the pdf fZ (z) of the sum of independent random variables Z = X1 + · · · + XN is the convolution of their pdf's fX1 (z), . . . , fX2 (z), the characteristic function of fZ (z) is the product of the individual functions as follows

fZ (z) = fX1 (z) ∗ fX2 (z) ∗ · · · ∗ fXN (z), (3.21)

ΦZ (ω) = ΦX1 (ω)ΦX2 (ω) . . . ΦXN (ω), where ∗ denotes convolution.

3.2.1 Higher-Order Moments If we consider the denition of the characteristic function in Equation (3.18), the exponent function inside the expectation can be expanded to a power series as follows

ΦX (ω) = E[e

jωX

]=E

∞ hX (jωX)n i n=0

= 1+

n!

=

∞ X (jω)n n=0

n!

mX n

jω X (jω)2 X (jω)3 X m + m2 + m3 + . . . , 1! 1 2! 3!

(3.22)

n X where mX n is the nth-order moment of X , i.e. mn = E[X ]. Now the nth-order moment of X can be obtained by calculating the nth-order derivative of ΦX (ω) with respect to ω at point ω = 0 as follows

mX n

¯ 1 dn ¯ = n n ΦX (ω)¯ . j dω ω=0

(3.23)

This implies that the knowledge of all the moments of X characterizes the pdf fX (x) completely. According to Equation (3.19) the Gaussian random variable X has innite number of moments due to the derivation rule of the exponential function.


32

If X[k] is a real stationary discrete-time signal and its moments up to order n exist, then [28, 29]

£ ¤ mX n (τ1 , τ2 , . . . , τn−1 ) , E X[k]X[k + τ1 ] . . . X[k + τn−1 ]

(3.24)

represents the nth-order moment function of {X[k]}, which depends only on the time dierences τ1 , τ2 , . . . , τn−1 , τi = 0, ±1, ±2, . . . for all i. It can be easily seen that the rst-order moment is the mean of the sequence and the second-order moment is the autocorrelation function as follows

µX , mX 1 , RX (τ ) , mX 2 (τ1 ).

(3.25)

3.2.2 Cumulants and Multi-Correlations If we take the natural logarithm of the characteristic function, we get the cumulative function ΨZ (ω) or the second characteristic function [12, 30]. The benet of the cumulative function comes from the fact that the product in the Equation (3.21) changes to a sum as follows

ΨZ (ω) , ln ΦZ (ω) = ln ΦX1 (ω) + ln ΦX2 (ω) + · · · + ln ΦXN (ω) = ΨX1 (ω) + ΨX2 (ω) + · · · + ΨXN (ω).

(3.26)

Similarly, the nth-order cumulant of a random variable X can be obtained by calculating the nth-order derivative of the cumulative function at ω = 0 as

cX n =

¯ 1 dn ¯ Ψ (ω) ¯ . X j n dω n ω=0

(3.27)

The nth-order cumulants of the Gaussian random variable X are identically zero for n > 2 because the cumulative function for X is a second-order polynomial of ω . From Equation (3.19), it follows that

ΨX (ω) = jµω −

σ2 2 ω . 2

(3.28)

The nth-order cumulant of a non-Gaussian stationary random signal, X[k], for n = 3, 4 can be written as [28] X G cX n (τ1 , τ2 , . . . , τn−1 ) = mn (τ1 , τ2 , . . . , τn−1 ) − mn (τ1 , τ2 , . . . , τn−1 ),

(3.29)


33

where the rst term mX n (τ1 , τ2 , . . . , τn−1 ) is the nth-order moment function of X[k] and mG (τ , τ , . . . , τ n−1 ) is the nth-order moment function of an equivalent Gausn 1 2 sian signal that has the same mean value and autocorrelation sequence as X[k]. G Obviously, if X[k] is Gaussian, mX n (τ1 , τ2 , . . . , τn−1 ) = mn (τ1 , τ2 , . . . , τn−1 ) and thus cX n (τ1 , τ2 , . . . , τn−1 ) = 0 for n = 3, 4. For a zero-mean process {X[k]}, the relationships between the moment and cumulant sequences can [28] be expressed as

£ ¤ X cX 1 = E X[k] = m1 = 0, £ ¤ X X X cX 2 (τ1 ) = E X[k]X[k + τ1 ] = m2 (τ1 ) = m2 (−τ1 ) = c2 (−τ1 ), £ ¤ X cX 3 (τ1 , τ2 ) = E X[k]X[k + τ1 ]X[k + τ2 ] = m3 (τ1 , τ2 ),

(3.30) (3.31) (3.32)

£ ¤ X X cX 4 (τ1 , τ2 , τ3 ) = E X[k]X[k + τ1 ]X[k + τ2 ]X[k + τ3 ] − c2 (τ1 )c2 (τ2 − τ3 ) X X X −cX 2 (τ2 )c2 (τ3 − τ1 ) − c2 (τ3 )c2 (τ1 − τ2 ) X X = mX 4 (τ1 , τ2 , τ3 ) − m2 (τ1 )m2 (τ3 − τ2 ) X X X −mX 2 (τ2 )m2 (τ3 − τ1 ) − m2 (τ3 )m2 (τ2 − τ1 ).

(3.33)

If the signal {X[k]} is zero-mean, it follows from Equations (3.31) and (3.32) that the second and third-order cumulants are identical to the second and third-order moments, respectively. Therefore the fourth-order cumulants are important if the random process is symmetrically distributed. By putting τ1 = τ2 = τ3 = 0 and assuming mX 1 = 0, we obtain very important results as follows

£ ¤ γ2X = E X 2 [k] = cX 2 (0),

(3.34)

£ ¤ γ3X = E X 3 [k] = cX 3 (0, 0),

(3.35)

£ ¤ γ4X = E X 4 [k] − 3(γ2X )2 = cX 4 (0, 0, 0).

(3.36)

These values describe the shape of the pdf of X[k]; i.e. γ2X is the variance and describes the width of the pdf, γ3X is the skewness and describes the symmetry, and γ4X is the kurtosis and describes the peakedness. 2 In Figure 3.2, there is a zero-mean Gaussian density with variance σX = 0.5 estimated from a random sequence of 200000 samples. Theoretical values of the skewness and kurtosis of the Gaussian pdf can be evaluated from Equation (3.28)


34

Single Gaussian 0.7

0.6

Probability density

0.5

0.4

0.3

0.2

0.1

0 −5

−4

−3 −2 −1 0 1 2 3 mean=−0.00, variance=0.50, skewness=−0.00, kurtosis=0.00

4

5

Figure 3.2: Gaussian pdf. and they should both be zero. The estimated values in Figure 3.2 are consistent with the theory. In Figure 3.3, there are two pdf's which both are mixtures of two Gaussians. These pdf's are

1 £ 2 2¤ f (x) = √ e(x+2) + e(x−2) 2 π

(3.37)

1 £ 2 2¤ f (x) = √ 2e(x+2) + e(x−2) 3 π

(3.38)

and

respectively. The pdf's are mixtures of two Gaussians with the same variance 2 σX = 0.5 and dierent means µ1 = −2 and µ2 = 2. The probability mass is distributed in the rst pdf 1:1 and in the second pdf 2 : 1. The theoretical values of mean, variance, skewness and kurtosis can be shown to be µX = 0, γ2X = 4 21 , 1 20 γ3X = 0, and γ4X = −32 for the rst pdf and µX = − 23 , γ2X = 4 18 , γ3X = 4 27 , and 26 X γ4 = −18 27 for the second pdf. The cumulants have several desirable properties that the moments do not possess. Due to these properties the cumulant can be treated as an operator, just like the expectation. These properties include [28] 1. Cumulants of scaled quantities, where the scale factors are non-random, equal the product of the scale factors times the cumulant of the unscaled quantities, i.e. if λi , i = 1, 2, . . . , n are constants and xi = X[k] are random variables then

cum(λ1 x1 , . . . , λn xn ) =

n ³Y i=1

´ λi cum(x1 , . . . , xn ).

(3.39)


35 Mixture of two Gaussians

0.7

0.6

0.6

0.5

0.5

Probability density

Probability density

Mixture of two Gaussians 0.7

0.4

0.3

0.4

0.3

0.2

0.2

0.1

0.1

0 −5

−4

−3 −2 −1 0 1 2 3 mean=−0.00, variance=4.50, skewness=−0.00, kurtosis=−31.98

4

5

0 −5

−4

−3 −2 −1 0 1 2 3 mean=−0.67, variance=4.06, skewness=4.73, kurtosis=−18.98

4

5

Figure 3.3: Probability densities with dierent statistics. 2. Cumulants are symmetric in their arguments, i.e.

cum(x1 , . . . , xn ) = cum(xi1 , . . . , xin ),

(3.40)

X where (i1 , . . . , in ) is a permutation of (1, . . . , n) e.g. cX 3 (τ1 , τ2 ) = c3 (τ2 , τ1 ).

3. Cumulants are additive in their arguments, i.e., the cumulants of sums equal the sums of cumulants. E.g. even when x and y are not statistically independent it follows that

cum(x + y, z1 , . . . , zn ) = cum(x, z1 , . . . , zn ) + cum(y, z1 , . . . , zn ).

(3.41)

4. Cumulants are blind to additive constants, i.e. if α is constant, then

cum(α + x1 , . . . , α + xn ) = cum(x1 , . . . , xn ).

(3.42)

5. Cumulants of a sum of statistically independent quantities equal the sum of the cumulants of the individual quantities, i.e. if the random variables {xi } are independent of the random variables {yi }, i = 1, 2, . . . , k , then

cum(x1 + y1 , . . . , xn + yn ) = cum(x1 , . . . , xn ) + cum(y1 , . . . , yn ).

(3.43)

If xi and yi are not independent, then the third property should be used to expand the left-hand side of the previous expression. 6. Cumulants of statistically independent random variables are zero, i.e. if {xi } is a statistically independent process

cum(x1 , . . . , xn ) = 0.

(3.44)


36

The p + q th-order multi-correlation of a complex-valued stationary random signal is dened [1] as follows

¡ CX,p+q,p (τ1 , . . . , τp+q−1 ) = cum X(t), X(t + τ1 ), . . . , X(t + τp−1 ), ¢ X ∗ (t − τp ), . . . , X ∗ (t − τp+q−1 ) .

(3.45)

In this notation p + q is the order of the multi-correlation whereas p represents the number of non-conjugated components. The properties of the multi-correlation are direct consequences of the properties of the cumulants described earlier.

3.2.3 Higher-Order Spectra Higher-order spectra (HOS) are multi-dimensional Fourier transforms of higherorder statistics. They are dened either in terms of cumulants or moments and their Fourier transforms are called cumulant spectra or moment spectra respectively [28]. Thus, the polyspectra are dened in terms of the cumulants as follows

CnX (ω1 , ω2 , . . . , ωn−1 )

=

∞ X τ1 =−∞

∞ X

···

cX n (τ1 , τ2 , . . . , τn−1 ) exp

©

−j

τn−1 =−∞

n−1 X

ª ωi τi .

i=1

(3.46)

The polyspectrum for n = 2 is the conventional power spectrum, for n = 3 it is called bispectrum and for n = 4 it is called trispectrum. The computational complexity of the polyspectra can be reduced by using many symmetries which arise from the properties of the cumulants. The symmetry regions of third-order cumulants and bispectrum are depicted in Figure 3.4. Knowing the third-order cumulant in one of the six sectors, I through VI, would enable us to nd the entire third-order cumulant sequence. Similarly, knowing the bispectrum in the triangular region ω2 ≥ 0, ω1 ≥ ω2 , ω1 + ω2 ≤ π is enough for a complete description of the bispectrum. The moment spectra MnX (ω1 , ω2 , . . . , ωn−1 ) for a nite energy signal {x[k]}, k = 0, ±1, ±2, . . . can be dened also using the Fourier transform as follows

X(ω) =

∞ X

x(k)e−jωk ,

(3.47)

k=−∞

M2X (ω) = X(ω)X ∗ (ω), M3X (ω1 , ω2 ) = X(ω1 )X(ω2 )X ∗ (ω1 + ω2 ), M4X (ω1 , ω2 , ω3 ) = X(ω1 )X(ω2 )X(ω3 )X ∗ (ω1 + ω2 + ω3 ).

(3.48) (3.49) (3.50)

CHAPTER 3. STATISTICAL TOOLS τ1

37 ω1

τ1= τ2 II

III

I τ2

ω2

IV V

VI

Figure 3.4: Symmetry regions of 3rd-order cumulants and bispectrum. The polycepstrum can be obtained similarly to the (power) cepstrum by taking the logarithm of the corresponding polyspectrum and inverse transforming the log-spectrum. Similarly to the polyspectrum, the polycepstrum is called power cepstrum for n = 2, bicepstrum for n = 3 and tricepstrum n = 4. A normalized higher-order spectrum or the nth-order coherency index is a function that combines the cumulant spectrum of order n with the power spectrum of a signal. The third and fourth-order coherency indices are called bicoherency and tricoherency, respectively, and are dened as

C3X (ω1 , ω2 )

P3X (ω1 , ω2 ) = p

C2X (ω1 )C2X (ω2 )C2X (ω1 + ω2 ) C4X (ω1 , ω2 , ω2 )

P4X (ω1 , ω2 , ω3 ) = p

(3.51)

,

C2X (ω1 )C2X (ω2 )C2X (ω3 )C2X (ω1 + ω2 + ω3 )

.

(3.52)

These functions are very useful in detection and characterization of nonlinearities and non-Gaussianity in time series and in discriminating linear processes from nonlinear ones. A signal is said to be a linear ¯ non-Gaussian ¯process of order n if ¯ ¯ the magnitude of the nth-order coherency, ¯Pnx (ω1 , . . . , ωn−1 )¯, is constant over all frequencies; otherwise, the signal is said to be a nonlinear process.

Chapter 4 Review of Modulation Classication The publicly available literature has very little information on the modulation classication. Yet there are several articles on the subject but the methods are mostly restricted to a few modulation types like the MPSK. The papers can be divided into two groups based on the classication approach: the maximum likelihood and pattern recognition. In this chapter we survey most of the published methods. Some of them are implemented in this thesis and compared to the other methods in the subsequent chapters.

4.1 Maximum Likelihood Approach In the maximum likelihood (ML) approach, the classication is viewed as a multiple hypothesis testing problem, where a hypothesis, Hi , is arbitrarily assigned to the ith modulation type of m possible types. The ML classication is based on the conditional pdf p(x|Hi ), i = 1, . . . , m, where x is the observation; e.g. a sampled phase component. If the observation sequence X[k], k = 1, . . . , n is independent and identically distributed (i.i.d), the likelihood function (LF), L(x|Hi ), can be expressed [37] as

p(x|Hi ) =

n Y

p(X[k]|Hi ) , L(x|Hi ).

(4.1)

l=1

The ML classier reports the j th modulation type based on the observation whenever L(x|Hj ) > L(x|Hi ), j 6= i; j, i = 1, . . . , m. If the likelihood function is exponential, the log-likelihood function (LLF) can be used due to the monotonicity of the exponent function. Often the expressions of the pdf's are approximate and assume prior information like the symbol rate and SNR. Hence, quasi-optimal rules are dened. The block diagram of a general maximum likelihood classier is given in Figure 4.1.

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 39 L(x |H1) Signal

SENSING (measuring)

Measurements

CHOOSE Type j THE LARGEST

x

L(x |Hm )

Figure 4.1: General maximum likelihood classier.

4.1.1 General Maximum Likelihood Methods Wei and Mendel developed a maximum likelihood method for classication of digital amplitude-phase modulations in [39]. The method is applicable to any constellation-based modulation types in an additive white Gaussian noise (AWGN) environment. The authors obtained the theoretical performance of the ML modulation classier that works under ideal conditions, and can serve as an upper bound of performance for any classier. All signal parameters were assumed to be available. Boiteau and Le Martret proposed a general maximum likelihood classier (GMLC) in [5] based on an approximation of the likelihood function. The authors derived equations of GMLC in the case of linear modulation and applied them to the MPSK/M'PSK classication where M > M 0 . The authors showed that their tests are a generalization of the previous methods using the ML approach discussed in Section 4.1.3. They deduced that the likelihood function of an observation given a reference can be closely approximated by a measure of the correlation between the empirical and the true temporal or spectral higher-order statistic. The GMLC provides a theoretical foundation for many empirical classication systems including those systems that exploit the cyclostationary property of the modulated signals.

4.1.2 MPSK Classier Based on the Exact Phase Distribution Yang and Liu proposed an asymptotic optimal algorithm for classifying the modulation type of general MPSK signals in [40]. Yang published the same results earlier with Soliman in [41] and in [43] with slightly dierent test statistics. The authors developed the classication algorithm by employing the exact phase distribution of a received MPSK signal, which was expressed in terms of the Fourier series expansion. The Fourier series coecients are illustrated in Figure 4.2 versus the signal-to-noise ratio (SNR). The authors showed a structure of this proposed classier for CW, BPSK, QPSK, and 8PSK. The maximum a posteriori (MAP) probability criterion was used to develop a multiple hypothesis classication rule. The MAP criterion reduced even-

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 40 0

10

−5

Coefficient value

10

−10

10

−15

10

−20

0 dB −2 dB −5 dB −8 dB −10 dB

10

−25

10

0

5

10

15

20

25

m

Figure 4.2: Coecients cm of the exact phase pdf. tually to a ML classier because the hypotheses were assumed equally likely. The performance of this classier was shown to be more eective than the earlier classiers in [41, 43]. The SNR was assumed to be known.

4.1.3 Classiers Based on the Likelihood Functions Quasi Log-Likelihood Ratio Classier Kim and Polydoros proposed a quasi log-likelihood ratio classier (qLLRC) for BPSK and QPSK in [20, 32] and compared it to the more traditional, ad-hoc techniques of the square-law classier (SLC) and the phase-based classier (PBC). The qLLRC was derived by approximating the likelihood functions of the phase modulated digital signals in white Gaussian noise. The authors showed that its performance is signicantly better than that of the intuitively designed PBC, or the conventional SLC. The capability of the qLLRC algorithm is, though, limited since it is only valid for low SNR (SNR< 0dB ). All signal parameters such as the carrier frequency, initial phase, symbol rate and SNR were assumed to be available. The classier can only be used to discriminate between the BPSK and QPSK signals.

M th -law Classier versus qM -Rule Hwang and Polydoros proposed a maximum likelihood classier based on the likelihood function of MPSK and MQAM signals in additive white Gaussian noise in [19]. The authors derived simplied versions of the LF for each modulation type denoted by the qM -statistic. The qM -classier can be interpreted as a synchronous, pulse-shape matched-lter. The performance of the qM -rule was compared against M th -law approaches like [34] reviewed in Section 4.2.2. The correct classication probability was approximated in a low SNR (SNR¿ 0dB ) with long observation time, i.e., N À 1 symbols. To achieve the same performance with the M th -law

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 41 classier additional gain in the SNR was shown to be more than 2dB . The qM -rule is only valid for low SNR (SNR< 0dB ) and all signal parameters such as the carrier frequency, initial phase, symbol rate and SNR were assumed to be available.

Average Log-Likelihood Ratio Classier Long, Chugg and Polydoros extended the low SNR methods to moderate and high SNR environments in [24]. The authors presented the Qm -rule based on the average log-likelihood ratio (ALLR) and an approximate expression for the pdf of the Qm -statistic was developed for medium and high SNR cases. The performance of the Qm -rule was evaluated in four dierent environments; e.g. in TV and CPFSK interference. The approximation of the ALLR demonstrated an ability to dramatically improve the performance compared to the qM -rule in [19]. The classier was developed for binary hypothesis testing and all signal parameters were assumed to be available.

Classication in Unknown ISI Environments Lay and Polydoros developed two classication techniques for digital modulated signals aected by an inter-symbol interference (ISI). The initial development of the classication tests have been derived assuming a known channel impulse response. The authors presented an average likelihood ratio test (ALRT) and a general likelihood ratio test (GLRT) and exploited a per-survivor processing (PSP) for the channel identication simultaneously. Pairwise classication tests for 16-ary digital modulations in known and unknown channels were simulated. The ALRT produced better performance than the GLRT but it required explicit knowledge of the signal power and noise variance for the channel. The GLRT only required the maximum likelihood estimate of the transmitted data and the computation of its decision statistics were considerably reduced through the use of the Viterbi algorithm. The simultaneous classication and channel estimation was found to be a very time consuming task and might skew the classication tests. The classiers were developed for binary hypothesis testing and all signal parameters other than the impulse response were assumed to be available.

Multiple Hypothesis Classier Chugg, Long and Polydoros extended the maximum likelihood modulation classication to include more than two hypothesized modulation types and to include an autonomous power estimation and threshold setting. A BPSK/QPSK/OQPSK classier was demonstrated, where OQPSK denotes the oset quadrature phase shift keying. The modulation classication was based on the average likelihood function (ALF), the threshold setting was based on quasi log-likelihood ratio test (qLLRT) and a maximum likelihood estimate for signal power was derived. All

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 42 other signal parameters were assumed to be available. It appeared that the most dicult aspect is to obtain a reliable power estimate when only the in-band measurements are available.

DFT of Phase Histogram and Modulus Based QAM Classication Schreyögg and Reichert presented a method to classify various QAM signal constellations by separately analyzing the DFT of the phase histogram and applying the knowledge about the distribution of the magnitude. The authors derived the likelihood functions and a rule to combine them for the classication was given. The pdf of the DFT bins of the phase histogram generated from the received symbols was used as the basis of the phase-based likelihood function. In the same way a modulus-based likelihood function was computed from the pdf of the constellations magnitude. The carrier frequency, symbol rate and SNR were assumed to be known. The performance of the classier was evaluated for the BPSK, QPSK, 8PSK, and for a few dierent QAM constellations. The likelihood functions could be easily derived for symmetric QAM modulation schemes.

4.1.4 Maximum Likelihood Classier for CPM The classication of the CPM signals according to their modulation indices was studied in [9, 17]. Huang and Polydoros proposed two classication rules for the CPM signals in low SNR in [17]. The classication rules are based on the loglikelihood functions (LLF) of the CPM signals in AWGN and can discriminate two single-index CPM signals with dierent modulation indices h1 and h2 . The rst rule e(h1 , h2 ) was equivalent to an energy comparator and the second c(h1 , h2 ) had a novel form. The second rule outperformed the rst one with short observations.

4.2 Pattern Recognition Approach A general pattern recognition system comprises of three parts; i.e. sensing, feature extraction, and decision procedures [38]. Each measurement, observation, or pattern vector x = (X[1], X[2], . . . , X[n])T describes a characteristic of the pattern or object. Often the pattern vectors contain redundant information. Thus, the dimensionality of the pattern space can be reduced. This reduction is often referred to as the preprocessing or feature extraction. The decision procedures consist of e.g. decision functions, distance functions, or neural networks. The block diagram of a general pattern recognition system is illustrated in Figure 4.3 [38].

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 43 Signal

SENSING (measuring)

Measurements x

FEATURE EXTRACTION (preprocessing)

Feature vectors

Pattern classes

DECISION

Figure 4.3: General pattern recognition system.

4.2.1 Envelope Based Methods Ratio of Dierent Envelope Statistics In [7], Chan and Gadbois proposed a classier based on the ratio (R) of the variance of the envelope to the square of the mean of the envelope. The authors derived the equations for R for four modulation types as a function of the carrier to noise ratio to set up the classication scheme. The classier determines the modulation type according to the domain R falls in. The performance of the classier for FM, AM, DSB, and SSB modulated signals was evaluated in a simulation. The authors concluded that the scheme is suitable for real time applications since the record length required for successful identication and the required computation time are short. Druckmann, Plotkin and Swamy studied and proposed a number of classication features in [10]. They presented a dierent method of the envelope extraction, which does not require the computation of the Hilbert transform. In addition to R in [7], the authors introduced four more features based on the analytic envelope and on the new envelope approximation to classify FM, AM, DSB, and SSB. The features employ ratios of dierent statistics of these two envelopes. The authors examined also the performance of a classication rule which utilizes two features and proposed a pair which gained a success rate of 99% in carrier-to-noise ratio (CNR) of 10dB . The proposed method for envelope extraction is nevertheless unsuitable for the complex envelope representation.

Deviations of Instantaneous Properties Azzouz and Nandi proposed nine features for the recognition of the analog and digital modulations in [3, 4, 27]. The features were derived from the signal spectrum and the instantaneous amplitude, frequency and phase. The features were used to classify analog AM, FM, DSB, USB, LSB, and digital 2ASK, 4ASK, 2PSK, 4PSK, 2FSK, 4FSK. Usually the classication of the 2PSK and 2ASK is impossible because the commonly used constellations of the 2PSK and 2ASK are identical. The authors used two dierent approaches in classifying the modulated signals. The rst approach was a decision theoretic tree classier where each feature was tested against a certain threshold value at a time. The success rate of the tree classier depends on the order of the features tested in these branches. The second approach was based on an articial neural network. In this approach, all features are considered simultaneously which should imply a better performance.

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 44

4.2.2 Higher-Order Methods Even Moment Based MPSK Classier Soliman and Hsue developed an automatic classication algorithm for carrier wave (CW) and M-ary PSK signals utilizing the statistical moments of the phase component of the received signal in [36]. Yang and Soliman also studied the momentbased classiers in [42]. In [36], the authors showed that for MPSK signals, even moments of the phase component are monotonic increasing functions of M. Based on this property, the authors formulated a general hypothesis test, developed a decision rule, and derived an analytic expression for the probability of the misclassication. At low carrier-to-noise ratio (CNR), the authors established that the eighth-moment is adequate for classifying the BPSK signals with a reasonable performance. Finally, the authors compared the performance of the suggested algorithm to the quasi log-likelihood ratio (qLLRC), square-law (SLC), and the phase-based (PBC) classiers [20]. The suggested algorithm was outperformed by the qLLRC algorithm at low CNR but had a comparable performance to the SLC and was better than the PBC algorithms.

The M th -law Based Classier Reichert [34] presented a classication method which utilizes dierent nonlinearities, i.e. squaring and fourth-power, applied to the pre-envelope of the digital modulated signal. The method exploited the dierences in the higher-order momentspaces of the discrete-time modulating process. These dierences contributed lines in the spectrum of the transformed signal, associated with the unknown carrier frequency fc and symbol rate fs . The spectral lines were detected by a periodogram analysis. Their existence, position and amplitude served as a robust feature for classifying the 2ASK, 2PSK, 4PSK, MSK, and 2FSK modulated signals. The author carried out a complete statistical analysis of the classication performance in terms of the probability of the detection and false alarm rate. The analytical performance gures were veried with simulated data. Minor defects of this method are the complexity of the periodogram analysis and its unsuitability for the complex envelope representation.

Cyclic Multi-correlation Based MPSK Classier In [25, 26], Marchand, Le Martret and Lacoume proposed a multiple hypothesis QAM modulation classier designed in the framework of decision theory. Same features were also introduced by Le Martret and Boiteau [22] in a slightly dierent framework. The proposed feature was composed of a combination between fourthorder and squared second-order cyclic temporal cumulants. The combination between the cumulants of dierent orders was intended to bypass the uncertainty

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 45 about the power of the signal of interest. The authors presented performance simulations in the context of 4QAM, 16QAM, and 64QAM classication. Comprehensive Monte-Carlo simulations for SNR values 5dB and 10dB were performed and the probability of the correct classication was examined. The classier exhibited poor performance for sample sizes less than 1024 symbols. The authors concluded that the feature introduced in their paper is the only one that can achieve QAM modulation classication in a cyclostationary context.

4.2.3 Other Methods Classication Exploiting Zero-Crossings Hsue and Soliman developed a classier using zero-crossing techniques in [15] and reported the same results a year later in [16]. The zero-crossing sampler had the advantage of providing accurate phase transition information over a wide dynamic frequency range. The authors estimated the zero-crossing variance, carrier-to-noise ratio (CNR), and carrier frequency by using these techniques. The phase dierence and zero-crossing interval histograms were used as the features for recognition of the CW, MPSK, and MFSK modulated signals. The obtained simulation results showed that a reasonable average probability of correct classication was achievable for CNR≥ 15dB . However, these techniques are unsuitable for the complex envelope representation.

Classication Based on Distance Functions Huo and Donoho developed a classication algorithm which used the counts of the signals falling into dierent parts of the signal plane as a feature in [18]. The feature was far more easier to compute and much faster than the likelihood methods and methods based on higher-order statistics. The optimal partition of the signal plane was derived for two candidate modulation types by maximizing the Hellinger distance in the multinomial distribution situation. The performance of the algorithm was veried with a classier implemented for the 4QAM versus 6PSK case. The proposed algorithm was thus dependent on the orientation of the symbols in the signal space and could only be used for binary classication.

Classication Based on the Modulation Model Assaleh, Farrell and Mammone proposed a new method for the modulation classication of the digital modulated signals [2]. The method utilized a signal representation known as the modulation model which is convenient for subsequent analysis such as the estimation of modulation parameters. The modulation model was formed via autoregressive spectrum modeling. It used the instantaneous frequency and band-width parameters as obtained from the roots of the autoregressive

CHAPTER 4. REVIEW OF MODULATION CLASSIFICATION 46 polynomial. In particular, the band-width parameter and the derivative of the instantaneous frequency were shown to provide excellent measures for information type and rate in addition to being noise robust. The method showed excellent preliminary results. The applicability of this method for the complex envelope representation was not studied.

Classication of CPM Chung and Polydoros proposed two envelope-based classiers for 2CPFSK signals [9]. The classier was developed for both the single and multi-index CPM signals. The classiers were based on the sum of the squared envelopes (SOSE). In the rst method, an appropriately adjusted threshold was able to classify a variety of candidate modulation sets. The second method could be used with innite index sets and was based on an approximate maximum likelihood estimate of the index pattern derived from SOSE. The binary SOSE scheme was compared to an LLF based method in [17]. The latter outperformed the SOSE scheme but all the methods worked reliably at very low SNR.

Chapter 5 Implemented Methods In this chapter, we present the methods implemented in this study and the setup used in the simulations. The implemented methods are described in more detail compared to the previous chapter. The pros and cons of these methods are discussed. We introduce a new feature derived in Appendix A, and its theoretical discrimination ability is discussed. The test signal generation is described as well.

5.1 Test Signal Generation for the Simulations Dierent modulated signals were generated to study some of the methods presented in this thesis. The modulation types were restricted to the types commonly used in radio communication. In the case of digital modulation, small constellations were used. To save the required amount of memory and to lower the sampling rate, the complex envelope was chosen as the signal representation. The modulation was carried out by using Matlab functions in Communications Toolbox. In the analog modulation, the modulating signal was a segment of narrow-band speech with sampling rate fs = 8kHz . First the modulating signal was re-sampled to fs = 100kHz and it was modulated with the carrier frequency fc = 25kHz . The speech signal was continuous but it cannot be expected to be stationary for a long period of time. Therefore, the results might be much dierent compared to those obtained with simulated modulating signals. The frequency deviation used in the frequency modulation was chosen to be f∆ = 20kHz . The modulation depth used in the amplitude modulation was chosen to be the maximum absolute value of the modulating signal to avoid any phase shifts. The analog modulation types produced for the simulations were AM, FM, DSB, and LSB. The digital modulated signals were obtained similarly to the analog modulation. Uniformly distributed discrete symbol sequences were modulated by the Matlab functions. Due to many source coding techniques, the digital bit sequences are uncorrelated. There is, though, some redundancy added by the channel coding but uniformly distributed random sequences of symbols were used as a modulating

CHAPTER 5. IMPLEMENTED METHODS

48

signal. The digital modulation types produced for the simulations were 2PSK, 4PSK, 8PSK, 2FSK, 4FSK, and 8FSK. A plain carrier wave (CW) was also used for the generality since it can be regarded as 1PSK or 1FSK. The symbol rate was chosen to be fd = 2500Hz because it should be an integer factor of the sampling rate. The phase states used in the generation of the 2PSK signals were φ = 0 and φ = π . For the 4PSK signals, the phase states were φ = 0, φ = π/2, φ = π , and φ = −π/2. The frequency deviation for the 2FSK signals was f∆ = 2500Hz , and f∆ = 1250Hz for the 4FSK signals. Additive white Gaussian noise (AWGN) was added to the modulated signals before conversion to the complex envelope representation. The signal-to-noise ratio (SNR) is dened as a ratio of the power of the signal to the power of the noise and it is usually expressed in decibels. The power of the signal s[k] can be estimated by its variance σs2 . First a normally distributed random sequence w[k] with zero-mean and unit-variance was generated by using a Matlab function. To obtain certain SNR, the noise sequence must be scaled by a constant,

s c=

σs2 , σw2 10SNR/10

(5.1)

where σw2 is the variance of the original noise sequence. The noisy signal is then r[k] = s[k] + cw[k]. The complex envelope was obtained by Hilbert transforming the noisy signal by using a Matlab function. The transformed signal was then multiplied by an exponential factor e−jωc k/fs , where the angular frequency of the carrier is ωc = 2πfc . The obtained signal was then re-sampled to fs = 50kHz , which performed the band-limitation simultaneously.

5.2 Implemented Methods 5.2.1 Ratio of Variance to Squared Mean This method for the analog modulation classication is reviewed in Section 4.2.1. The received signal is modeled in the identication scheme [7] as

r(t) = x(t) + w(t),

(5.2)

where the signal x(t) has an unknown modulation type to be determined and w(t) is a stationary, Gaussian white noise sequence with variance σw2 and uncorrelated with x(t). The analytic envelope used in this method is dened as

ξ(t) = r2 (t) + r˜2 (t),

(5.3)


49

where r˜(t) is the Hilbert transform of the received signal r(t). The identication scheme performs the following steps. It rst calculates the analytic envelope of r(t), then the mean value, µξ , and variance, σξ2 , of the envelope, and nally the ratio R = σξ2 /µ2ξ . The value of R determines the modulation type present in x(t). The authors derived the theoretical values of the ratio R by means of the carrier-to2 noise power r = A2 /2σw2 and the modulating-signal-to-noise ratio q = A2 σm /2σw2 , 2 where σm is the variance of the modulating signal. The theoretical values can be expressed as

FM:

R=

1 + 2r , (1 + r)2

(5.4)

AM:

R=

1 + 2r + 2q + 4qr + 2q 2 , (1 + r + q)2

(5.5)

DSB:

R=

1 + 2q + 2q 2 , (1 + q)2

(5.6)

SSB:

R = 1.

(5.7)

It follows from Equation (5.4) that the value of R tends to zero as r grows larger. Similarly the value of R in Equation (5.6) tends to 2 as q tends to innity. The values of R for the FM, AM, DSB, and SSB signals are plotted against various carrier-to-noise ratios r in Figure 5.1. The variance of the modulating signal used 2 in the gure is σm = 0.25. 2

FM AM DSB SSB

1.8

1.6

1.4

Ratio

1.2

1

0.8

0.6

0.4

0.2

0

0

10

20

30 CNR in dB

40

50

60

Figure 5.1: Theoretical values of the ratio R.


50

5.2.2 Deviations in Instantaneous Properties Another approach for the analog modulation classication is introduced in Section 4.2.1. Key features γmax , σap , σdp , and P are used to classify analog modulated signals. The key feature γmax represents the maximum value of the spectral power density of the normalized centered instantaneous amplitude acn [k] of the received signal. The value of γmax can be obtained by the maximum value of the DFT sequence of acn [k] as follows

γmax

s −1 ¯ NX ¯2 ¯ ¯2 ¯ ¯ ¯ ¯ = max DF T [acn [k]] /Ns = max ¯ acn [k]e−jkn2π/Ns ¯ /Ns ,

n

(5.8)

k=0

where Ns is the number of the samples in a segment. By the Parseval's theorem the power spectral density is equivalent to the square of the absolute values of the DFT sequence. Another key feature σap is the standard deviation of the absolute value of the centered nonlinear component of the instantaneous phase evaluated over the nonweak intervals of a signal segment. The non-weak intervals are dened as the indices for which the normalized amplitude exceeds a certain threshold an [k] > at . The estimation of the instantaneous phase is very sensitive to noise below that threshold. The value of σap can be obtained as follows

σap

s ³ ´ ³1 X 1 = φ2N L [k] − C C an [k]>at

X

´2 |φN L [k]| ,

(5.9)

an [k]>at

where C is the number of the samples φN L [k] for which an [k] > at . The key feature σdp is the standard deviation of the centered nonlinear component of the direct instantaneous phase, evaluated over the non-weak intervals of a signal segment. Similarly, σdp is dened by

σdp

s ³ ´ ³1 X 1 2 = φN L [k] − C C an [k]>at

X

´2

φN L [k] .

(5.10)

an [k]>at

The key feature P measures the spectrum symmetry about the carrier frequency and is calculated by the dierence of the power in the upper and lower-sidebands normalized by the total power. In the case of the complex envelope signal, the carrier frequency fc = 0. The key feature P can be expressed as

P =

PL − PU , PL + PU

(5.11)


51

where

PL =

fcn X

2fcn +1 2

|X[n]| ,

and

X

PU =

n=1

|X[n]|2 .

(5.12)

n=fcn +2

In Equation (5.11), X[n] is the DFT sequence of the signal x[k] and fcn + 1 is the sample number corresponding to the carrier frequency as follows

X(n) =

N s −1 X

x[k]e−jkn2π/Ns ,

and

fcn =

k=0

f c Ns − 1. fs

(5.13)

Key features γmax , σap , σdp , σaa , and σaf are used in classication of the digital modulated signals. Key feature σaa is the standard deviation of the absolute value of the normalized centered instantaneous amplitude of a signal segment. Similarly, σaa is dened by

σaa

v u ³ Ns Ns ´2 ´ ³ 1 X u 1 X t 2 a [k] − |acn [k]| . = Ns k=1 cn Ns k=1

(5.14)

The key feature σaf is the standard deviation of the absolute value of the normalized centered instantaneous frequency, evaluated over the non-weak intervals of a signal segment. Similarly, σaf is dened by

σaf

s ³ ´ ³1 X 1 fN2 [k] − = C C an [k]>at

X

´2 |fN [k]| .

(5.15)

an [k]>at

All the previous features and three new features are used to classify both the analog and digital modulated signals. The key feature σa is the standard deviation of the normalized centered instantaneous amplitude, evaluated over the non-weak intervals of a signal segment. It is dened by

s ³ ´ ³1 X 1 2 acn [k] − σa = C C an [k]>at

X

´2 acn [k] .

(5.16)

an [k]>at

The key feature µa42 is the kurtosis of the normalized centered instantaneous amplitude and is dened by

µa42 = ¡

E[a4cn (t)] ¢2 , E[a2cn (t)]

(5.17)

which is another denition of the kurtosis than the one given in Section 3.2.2. This feature was used to discriminate the analog and digital amplitude modulations AM,


52

2ASK, and 4ASK. In the simulation part, we have not used any ASK modulated signals. Therefore this feature is not used in the implementation. The key feature µf42 is the kurtosis of the normalized centered instantaneous frequency and is dened by

µf42 = ¡

E[fN4 (t)] ¢2 . E[fN2 (t)]

(5.18)

This feature was used to discriminate FM, 2FSK, and 4FSK.

5.2.3 Even Moments of MPSK Signals This method is presented in Section 4.2.2. In [36], the phase of the received signal is extracted by the methods given in Section 2.1. For a signal contaminated by AWGN, the extracted phase can be represented as

φα [k] = θα [k] + v[k],

−π < φα [k] ≤ π,

(5.19)

where θα [k] is the sampled phase component of the transmitted MPSK signal, and v[k] is the contribution of the noise and any other measurement error. Without loss of generality, φα [k] and θα [k] are assumed independent and identically distributed (i.i.d.) with zero-means. The number of the symbols is given by M = 2α , α = 0, 1, 2, . . . , where α = 0 corresponds to a CW signal. For the CW signal (α = 0), the pdf of the phase is given as

e−γ fφ (φ0 ) = + 2π

r

h p i γ 2 cos(φ0 )e−γ sin (φ0 ) Q − 2γ cos(φ0 ) , π

−π < φ0 ≤ π,

where

1 Q[x] = √ 2π

Z

∞

e−y

2 /2

dy

x

and

γ=

A2 , 2σw2

(5.20)

and A is the amplitude of the transmitted CW signal, and σw2 is the noise variance. Due to the complicated nature of Equation (5.20), it can be approximated by the Tikhonov pdf given as

fφ (φ0 ) '

exp[2γ cos(φ0 )] , 2πI0 [2γ]

−π < φ0 ≤ π,

(5.21)


53

where I0 [·] is the zero-order modied Bessel function of the rst kind. The pdf of θα can be written as α

2 1 X fθ (y; α) = α δ[y − ηk (α)], 2 k=1

(5.22)

where ηk (α) is given as

ηk (α) =

2k − 2α − 1 , 2α

k = 1, 2, . . . , 2α ,

α = 0, . . . , log2 M.

(5.23)

By combining Equations (5.19), (5.21), and (5.22) we obtain the pdf fφ (y; α) as a sum of non-central Tikhonov functions, which can be expressed as α

2 1 X exp[2γ cos(y − ηk (α))] fφ (y; α) = α . 2 k=1 2πI0 [2γ]

(5.24)

The approximations of fφ (y; α) for α = 0, 1, 2, 3 obtained from the histograms of the simulated signals are given in Figure 5.2. The number of the peaks in the gures indicate the number of the signal phase states. As SNR decreases, the peaks smear o and nally the pdf approaches that of a uniformly distributed random variable. As shown in Figure 5.2, for large α, fφ (y; α) is atter than those with small α and fφ (y; α) will approach 1/2π as SNR → −∞ dB or α → ∞. In the presence of noise, the nth-order moment of φα is given in Equation (5.25). The authors proved a theorem which states that the even moments mn (α) of the phase of the MPSK signals are monotonic increasing functions of α where α = 0, 1, . . . , log2 M .

Z

π

mn (α) =

y n fφ (y; α)dy

(5.25)

−pi

=

 1  2α+1 πI

0 [2γ]

 0,

P2α i=1−π

Rπ −π

y n exp[2γ cos(y − ηi (α))]dy, n even n odd

Figure 5.3 shows the moments of MPSK signals at SNR = 8 dB. The values are given in a logarithmic scale. The gure shows also that as α grows the dierence mn (α + 1) − mn (α) gets smaller and higher-order moments may be needed to discriminate between the signals with large M.

2

2

1.8

1.8

1.6

1.6

1.4

1.4 Probability density

Probability density


1.2

1

0.8

1

0.8

0.6

0.4

0.4

0

0.2

−3

−2

−1

0 Phase angle of CW

1

2

0

3

2

2

1.8

1.8

1.6

1.6

1.4

1.4 Probability density

Probability density

1.2

0.6

0.2

1.2

1

0.8

0.4

0.2

0.2

−1

0 Phase angle of PSK4

1

2

3

−1


1

2

3

−3

−2

−1


1

2

3

1

0.6

−2

−2

0.8

0.4

−3

−3

1.2

0.6

0

54

0

Figure 5.2: Pdf of CW and MPSK signals at SNR = 10 dB. 4

10

PSK8 PSK4 PSK2 CW

3

10

2

Moment value

10

1

10

0

10

−1

10

−2

10

2

3

4

5

6 Moment order

7

8

9

10

Figure 5.3: Moments of MPSK signals at SNR = 8 dB. The suggested algorithm uses estimates of the ensemble moments. Convenient estimates are the unbiased sample averages of the even powers of the extracted phase. They can be expressed as L

m ˆ n (α) =

1X n φ [k]. L i=1 α

(5.26)

By the central limit theorem, it can be shown that the pdf of m ˆ n (α) approaches


55

that of a Gaussian density with µn (α) = mn (α) as L increases. Therefore the random variables m ˆ n (α) have means that are monotonic increasing functions of α and can be used as a discriminating feature to classify the MPSK signals. A signicant implementation issue arises from the orientation of the MPSK constellation. For example if the alphabet for the 2PSK consists of the phase states φ = 0 and φ = π , the distribution of the noisy signal would be far away from the one shown in Figure 5.2. This problem might be solved by examining the behavior of the ensemble moments after dierent rotations of the constellation; i.e. by multiplying the signal by a complex exponential ejπ/M , M = 2, 4, 8, . . . before extracting the phase component.

5.2.4 Time-Average of Complex Envelope MFSK Process First and second-order statistics of digital modulated signals are studied in Appendix A. The modulated signals are examined as complex envelope random processes where the discrete symbol sequence is unknown. The modulating symbol sequence is assumed to be i.i.d. with a uniform probability mass function (PMF). The mean and autocorrelation functions are derived for the carrier wave (CW), the M-ary amplitude shift keying (MASK), the M-ary phase shift keying (MPSK), and the M-ary frequency shift keying (MFSK). All these modulation types are established to be wide-sense cyclostationary. Therefore the autocorrelation function can be expanded to its Fourier series; i.e. cyclic autocorrelation function (CAF). The cyclic autocorrelation functions for all the modulation types mentioned are derived as well. The mean functions, µC (t), are shown to be identically zero for the MASK and MPSK processes and identically unity for the CW process whereas for the MFSK process, the mean function is found to depend on the number of the symbols, M . Therefore the time-average or the expectation of the mean function with respect to time can be regarded as a useful feature for separating the CW, MASK/MPSK, and MFSK with dierent number of symbols. Let the received signal contaminated by zero-mean additive white Gaussian noise be as follows

r(t) = xc (t) + w(t),

(5.27)

where xc (t) is the original modulated signal and w(t) is the noise. The analytic representation of r(t) can be expressed as

¡ ¢ z(t) = xc (t) + w(t) + j x˜c (t) + w(t) ˜ = zx (t) + zw (t),

(5.28)


56

where zx (t) is the analytic representation of the original modulated signal xc (t) and zw (t) is the analytic representation of the noise w(t). In Equation (5.28), we have used the linearity property of the Hilbert transform. The complex envelope representation of r(t) can be expressed as

£ ¤ c(t) = z(t)e−jωc t = zx (t) + zw (t) e−jωc t .

(5.29)

The time-average of the complex envelope representation of the received signal c(t) can be divided into the time-average of the modulated signal component and the time-average of the noise component. The time-average of the noise component will be zero due to the properties of the AWGN, Hilbert transform, and multiplication by a complex exponential [31, 30]. The time-average of the complex envelope of r(t) depends only on the time-average of the original modulated signal. Therefore the time-averages derived for the digital modulated signals in Appendix A can be used as a distinctive feature. These features are summarized in Table 5.1. The ensemble estimate of the time-average is an attractive feature due to its robustness against the zero-mean noise and fast calculation. The number of symbols required for consistent recognition is studied by simulations in the next chapter. The autocorrelation function, RC (t1 , t2 ), is also identically zero for the MPSK process when M = 4, 8, 16, . . . and unity for the CW process whereas for the other modulation types mentioned, the autocorrelation function depends on M . The cyclic autocorrelations are found to exhibit similar characteristics. Further analysis is needed to exploit the CAF in the modulation recognition. The features of the autocorrelation functions are summarized in Table 5.1, as well. Table 5.1: Summary of rst and second-order statistics. CW MASK MPSK MFSK µC 1 0 0 1/M RC (t1 , t2 ) 1 depends on M 0 for M = 4, 8, 16, . . . depends on M

Chapter 6 Simulation Results In this chapter, the methods reviewed in the previous chapter are simulated. The results are given as gures where their discrimination eciency can be seen. Also the number of the symbols needed for the discrimination is studied in the case of the features used with digital modulation types. In the last section, we discuss about the proper features for our objectives.

6.1 Results 6.1.1 Ratio of Variance to Squared Mean The R values of the analog modulated signals against the signal-to-noise ratio (SNR) calculated over a segment of 51200 samples are depicted on the left hand side of Figure 6.1. The order of dierent modulation types meets the theoretical values reviewed in Section 5.2.1. The actual values are though quite dierent from those illustrated in Figure 5.1. This is a consequence of the assumptions made in the derivation of the theoretical values. E.g. it is clear that the variance of the modulating signal is not a constant. The modulation types can be separated pairwise by a single straight line down to about 15dB . The behavior of the feature against the number of the samples at SNR 15dB can be seen on the right hand side of Figure 6.1. Again the order of the modulation types seems to be correct with a small number of samples but the variance is large. The values converge after 1000 samples and 2048 samples used in the original paper [7] is adequate.

6.1.2 Deviations in Instantaneous Properties The features in [3] are simulated for the analog and digital modulation types. The segment size used was chosen to be 2048; same as in [4, 3]. The key feature

CHAPTER 6. SIMULATION RESULTS

58

R values

R values at SNR 15dB

2

2.5

AM FM DSB LSB

1.8

AM FM DSB LSB

1.6

2

1.4

1.2

1.5

1

0.8

1

0.6

0.4

0.5

0.2

0

0

5

10

15

20 25 Signal to noise ratio

30

35

40

0

200

400

600

800 1000 1200 The number of samples

1400

1600

1800

2000

Figure 6.1: R values versus SNR and number of samples. values are averaged from 25 segments, which is justied by the majority logic rule applied in the classication procedure. The key features γmax , σdp , σap , and P are presented in Figures 6.2 and 6.3 against SNR. The key feature γmax represents the spectral power density maximum and should be zero for the frequency modulation and large for all types of amplitude modulation. Due to the complex envelope representation, the contribution of the carrier in the amplitude modulation is small when the instantaneous amplitude is normalized and centered. Therefore the value of γmax is small as can be seen on the left hand side of Figure 6.2. The key feature P measures the spectrum symmetry and is presented on the right hand side of Figure 6.2. As seen in Section 2.2, the amplitude modulated (AM) and double-sideband modulated (DSB) signals have symmetric whereas the singlesideband modulated (SSB) and frequency modulated (FM) signals have asymmetric spectra. The spectrum symmetry measure is largest for the SSB signals due to the absence of the other half of the spectrum about the carrier. The FM signals have always information on the both sides of the carrier frequency but it is not symmetrically distributed. Spectrum symmetry measure

Spectral power density maximum

1

250

AM FM DSB LSB

AM FM DSB LSB

0.9

0.8

200

0.7

0.6

150

0.5

0.4

100

0.3

0.2

50

0.1

0

5

10

15


30

35

40

0

5

10

15


30

35

40

Figure 6.2: Instantaneous amplitude of analog modulation. The key feature σdp measures the standard deviation of the centered nonlinear


59

component of the direct instantaneous phase and the key feature σap measures the standard deviation of the absolute instantaneous phase. These features are plotted against the SNR in Figure 6.3. Due to the selection of the modulation depth, the amplitude modulation has no phase information at all. There are though some phase shifts in all the modulation types when the SNR is low; i.e. the extraction of the phase component becomes very imprecise at a low SNR. For example the phase of the DSB signal varies between −π and π when it should be π . Therefore the DSB signals have the largest values of both σdp and σap . Standard deviation of absolute phase

Standard deviation of direct phase

1.6

2.5

AM FM DSB LSB

1.4

AM FM DSB LSB 2

1.2

1

1.5

0.8

1

0.6

0.4 0.5 0.2

0

5

10

15


30

35

40

0

5

10

15


30

35

40

Figure 6.3: Instantaneous phase of analog modulation. The normalized instantaneous amplitude threshold at used in the calculation of the key features σdp and σap was chosen to be at = 1 in Figure 6.3. The optimal value found via the classication performance in [4] was 0.80 ≤ at ≤ 1.05 for the analog modulation. The inuence of the amplitude threshold near value at = 1 and the SNR 15dB is illustrated in Figure 6.4 for the key features σdp and σap . It is clear that the amplitude threshold has no signicant inuence on these key features on average. Standard deviation of direct phase

Standard deviation of absolute phase

2.5

1.6

AM FM DSB LSB

AM FM DSB LSB

1.4

2 1.2

1

1.5

0.8

1

0.6

0.4 0.5 0.2

0 0.8

0.85

0.9

0.95

1 1.05 Amplitude threshold

1.1

1.15

1.2

0 0.8

0.85

0.9

0.95


1.1

1.15

1.2

Figure 6.4: Inuence of amplitude threshold on analog modulation. The key features γmax , σdp , σap , σaa , σaf , and µf42 for the digital modulated signals are presented in Figures 6.5, 6.6, 6.7, and 6.8. The key feature γmax on the left hand side of Figure 6.5 shows how the band-limitation has the largest eect on the


60

quadrature phase shift keyed (QPSK or 4PSK) signals. The band-limitation with respect to the symbol rate is not strict enough to cause any signicant variations to the instantaneous amplitude of the binary phase shift keyed (BPSK or 2PSK) signals. Clearly the frequency shift keyed (FSK) signals have very small values of the spectral power density maximum. A plain carrier wave (CW) is also presented in Figure 6.5. Obviously γmax is close to zero because the only maximum is at the carrier frequency which is considered as zero. The key feature σaa on the right hand side of Figure 6.5 measures the standard deviation of the absolute value of the normalized instantaneous amplitude. As in the previous paragraph, the 4PSK has more deviation in the instantaneous amplitude due to the band-limitation. Otherwise the robustness of this feature is quite poor. Spectral power density maximum

Standard deviation of absolute amplitude

3.5

0.2

CW 2PSK 4PSK 2FSK 4FSK

3

2PSK 4PSK 2FSK 4FSK

0.18

0.16

2.5

0.14

0.12

2

0.1

1.5 0.08

0.06

1

0.04

0.5 0.02

0

0

5

10

15 Signal to noise ratio

20

25

30

0 10

12

14

16

18 20 22 Signal to noise ratio

24

26

28

30

Figure 6.5: Instantaneous amplitude of digital modulation. The angle modulated signals have information in the phase component as seen in Figure 6.6. The key feature σdp on the right hand side has larger values for the 2PSK than 4PSK due to the oscillation of the instantaneous phase between −π and π caused by the noise. This problem is similar to the one which evolves in the calculation of the even moments in the next section. Therefore the behavior of these features depends heavily on the orientation of the constellation as depicted in Figure 6.7. The features in Figure 6.7 were calculated after multiplying the 2PSK signals by a complex exponential ejπ/2 and the 4PSK signals by ejπ/4 . The CW signals do not carry any phase information other than the noise and therefore σdp is close to zero. The FSK signals have also larger deviation in the direct phase component compared to the absolute phase component because the former varies between −π and π whereas the latter varies between 0 and π . The key feature σap on the left hand side of Figure 6.6 measures the standard deviation of the absolute phase component which gives about π/2 for the 2PSK due to its properties. If the oscillations mentioned above are prevented; i.e. the constellation of the PSK signals is dierent, σap should be near zero for the phase shift keyed signals as seen in Figure 6.7. The key features, σaf and µf42 , based on the instantaneous frequency are illustrated


61



1.6

2.5

2PSK 4PSK 2FSK 4FSK

1.5


2 1.4

1.3

1.5

1.2

1

1.1

1 0.5 0.9

0.8

0

5

10


20

25

30

0

0

5

10


20

25

30

Figure 6.6: Instantaneous phase of digital modulation. in Figure 6.8. The key feature σaf measures the standard deviation of the absolute value of the normalized centered instantaneous frequency which is lowest for the 2FSK signals. The large value for the other modulation types is due to the impulses with a large amplitude as derived in Section 2.3. This key feature discriminates modulation types poorly. The key feature µf42 on the right hand side of Figure 6.8 is the kurtosis of the normalized centered instantaneous frequency and is used to discriminate between dierent frequency modulated signals. As kurtosis measures the peakedness of the distribution, µf42 should be small for the FM signals and large for the FSK signals depending on the frequency deviation and the number of the symbols. However, due to the dierentiation, there will be impulses with a large amplitude at the symbol transitions which broadens the distribution of the instantaneous frequency. This key feature performs poorly in discriminating the FM and 2FSK signals. Again the key features above are calculated using the amplitude threshold value at = 1. For the digital modulated signals, the optimal threshold value was found in [4] to be 0.99 ≤ at ≤ 1.05. The inuence of dierent amplitude thresholds near at = 1 for the key features σdp , σap , and σdp is illustrated in Figure 6.9. Obviously the inuence is not very signicant. Standard deviation of absolute phase


1.2

2

2PSK 4PSK 2FSK 4FSK

1


1.8

1.6

1.4 0.8 1.2

0.6

1

0.8 0.4 0.6

0.4 0.2 0.2

0

0

5

10


20

25

30

0

0

5

10


20

25

30

Figure 6.7: Instantaneous phase of digital modulation with dierent constellations.

CHAPTER 6. SIMULATION RESULTS Kurtosis of the instantaneous frequency

Standard deviation of absolute frequency

30

6000

FM 2FSK 4FSK

2PSK 4PSK 2FSK 4FSK

5000

25

4000

20

3000

15

2000

10

1000

5

0

0

5

10


62

20

25

30

0

0

5

10


20

25

30

Figure 6.8: Instantaneous frequency of digital modulation.

6.1.3 Even Moments of MPSK Signals In this section, we have assumed that the constellation of the received phase shift keyed signal has a correct orientation or it has been corrected in advance. The values of the ensemble moments with the orders n = 2, 4, 6, 8 are shown in Figure 6.10. The second-order moment on the upper left hand side corner can reliably discriminate the carrier wave from the binary phase shift keyed signals whereas more reliable discrimination of larger symbol sets are obtained by the higher-order Standard deviation of direct phase


2.2

1.6

2PSK 4PSK 2FSK 4FSK

2.1

2PSK 4PSK 2FSK 4FSK

1.5

2 1.4 1.9 1.3

1.8

1.7

1.2

1.6

1.1

1.5 1 1.4 0.9

1.3

1.2 0.8

0.85

0.9

0.95


1.1

1.15

4

3

1.2

0.8 0.8

0.85

0.9

0.95


1.1

1.15

1.2

Standard deviation of absolute frequency

x 10

2PSK 4PSK 2FSK 4FSK

2.5

2

1.5

1

0.5

0 0.8

0.85

0.9

0.95


1.1

1.15

1.2

Figure 6.9: Inuence of amplitude threshold on digital modulation.


63

moments. The CW, 2PSK, and 4PSK signals can be discriminated at the SNR less than 0dB by using the second and fourth-order moments. The fourth moment

The second moment

25

4

CW 2PSK 4PSK 8PSK

CW 2PSK 4PSK 8PSK

3.5

20 3

2.5

15

2

10 1.5

1

5 0.5

0

0

5

10


20

25

30

0

0

5

10

The sixth moment


20

25

30

The eighth moment 1200

150

CW 2PSK 4PSK 8PSK

CW 2PSK 4PSK 8PSK

1000

800

100

600

400

50

200

0

0

5

10


20

25

30

0

0

5

10


20

25

30

Figure 6.10: Even moments of PSK signal against SNR. The inuence of the number of the symbols in a segment to be classied is presented in Figure 6.11. The discrimination of the carrier wave seems to work with few symbols in a segment but good performance with larger symbol sets requires at least 20 symbols in a segment. In this case the segment size is 400 samples because one symbol takes 20 samples with our choice of the sampling frequency and the symbol rate. Therefore previously used segment size of 2048 samples is more than adequate for this feature to perform correctly.

6.1.4 Time-Average of Complex Envelope MFSK Process The performance of the new feature proposed in Section 5.2.4 can be seen in Figure 6.12. There are 100 symbols in a segment on the left hand side of the gure; i.e. the size of the segment is 2000 samples. The values are close to the theoretical values and tend to these values due to the law of large numbers (LLN). The convergence of the time-average against the number of the symbols is shown on the right hand side of Figure 6.12. Again the carrier wave can be discriminated from the other modulation types with very small segments. As the size of the constellation grows, more symbols in a segment is required for reliable classication. In this example the correct order evolves after about 35 symbols.


64

The second moment at SNR 8dB

The fourth moment at SNR 8dB

6

50

CW 2PSK 4PSK 8PSK

5

CW 2PSK 4PSK 8PSK

45

40

35

4

30

3

25

20

2 15

10

1 5

0

2

4

6

8

10 12 The number symbols

14

16

18

20

0

2

4

6

8


The sixth moment at SNR 8dB

14

16

18

20

The eighth moment at SNR 8dB

300

2000

CW 2PSK 4PSK 8PSK

250

CW 2PSK 4PSK 8PSK

1800

1600

1400

200

1200

150

1000

800

100 600

400

50 200

0

2

4

6

8


14

16

18

0

20

2

4

6

8


14

16

18

20

Figure 6.11: Even moments against number of symbols at SNR 8dB.

6.2 Discussion on Features The above results can be used to decide which features are useful for the actual decision procedures. The most interesting features are gathered in Tables 6.1 and 6.2 where the modulation types are considered pairwise. The discriminating features are presented in the intersections of every pair. The key features γmax , σdp , and P are presented in Section 5.2.2, m4 refers to the fourth-order moment of the instantaneous phase and is presented in Section 5.2.3, and µC refers to the Mean of digital modulated signals, 100 symbols

Mean against the number of symbols

1.5

1.5

CW 2FSK 4FSK 8FSK MPSK

CW 2FSK 4FSK 8FSK MPSK

1

1

0.5

0.5

0

0 0

5

10


20

25

30

5

10

15

20

25 30 35 The number of symbols

40

45

50

55

Figure 6.12: Time-average against SNR and number of symbols.

60


65

time-average of the complex envelope and is presented in Section 5.2.4.

FM DSB SSB CW 2PSK 4PSK 2FSK 4FSK

Table 6.1: Analog modulation types versus all. AM FM DSB SSB γmax , σdp , P γmax , σdp γmax , σdp , P γmax , σdp , P γmax , P P γmax σdp , P γmax , σdp γmax , σdp , P γmax , σdp σdp γmax γmax γmax , σdp σdp γmax γmax γmax , σdp µC γmax γmax γmax , σdp µC γmax γmax

All the modulation types can be discriminated pairwise from the other types. Therefore the most intuitive classication rule is a decision tree, where the features are tested in the branches against a certain threshold value and some of the candidate types can be discarded according to the result of this comparison [3]. E.g. the key feature γmax divides the modulation types into two groups: a group of the AM, DSB, and SSB signals, and a group of the FM, CW, 2PSK, 4PSK, 2FSK, and 4FSK. These groups can be further divided into subgroups, until only one type is left.

2PSK 4PSK 2FSK 4FSK

Table 6.2: Digital modulation types versus all. CW 2PSK 4PSK 2FSK 4FSK σdp , µC σdp , µC m4 σdp , µC µC µC σdp , µC µC µC µC -

Another classication rule is based on the feature-space, spanned by more than one feature considered simultaneously [38]. A set of cluster centers in the featurespace can be obtained by several algorithms such as Maximin-Distance, K-means, or Isodata. After training, each cluster center represents a certain modulation type. All the feature values, which represent a point in the feature-space, are calculated for the received signal. The decision is done according to the location of this point relative to the cluster centers using either decision functions or distance functions. All the feature values can also be used as an input to a neural network, which outputs the most probable modulation type [3]. In this approach, all the features are considered simultaneously as well. The decision procedures are out of the scope of this study and it requires much more attention to develop an automatic modulation recognizer.

Chapter 7 Conclusions In this study, several features for modulation classication were studied. The relevant characteristics of the communication signals and statistical tools were presented in the second and third chapters. A literature review of the previous methods was carried out and the most promising methods were surveyed in more detail. Further study of rst and second-order statistics including cyclostationary statistics was done for digital modulated signals. During these derivations a novel feature for the classication of the frequency shift keyed signals was found. The most interesting key features for pattern recognition purposes were selected according to their applicability for the minimal signal representation and for the classication of radio communication signals. The modulation types to be classied were selected emphasizing on these requirements. Therefore, the modulation types with a constant envelope and small symbol sets were used in the simulations. At the moment, an attractive application area would be software radio and other recongurable communication systems as real-time operating systems are being developed for signal processors. The pros and cons of the prospective features were discussed, as well. The discrimination performance of the potential features was studied by simulations with articially generated communication signals contaminated by additive white Gaussian noise. Also the robustness against noise was one of the criteria in the selection of the most interesting features. Simulations were carried out to study this behavior in dierent signal-to-noise ratio conditions. Also the inuence of the sample set size was studied. The proposed feature showed very promising results in a zero-mean noise environment. It also has many desirable properties that are required from reliable statistical estimates such as consistency. The cyclostationary statistics of the digital modulated signals might yield promising features. In comparison to higher-order statistics, they possess some desirable properties; e.g. lower variance. Due to their complicated appearance shown in this study, the features derived in the cyclostationary context will need further analysis. Also the inuence of dierent constellations on the classication of the phase shift keyed signals and its compensation would require more attention since

CHAPTER 7. CONCLUSIONS

67

the reviewed methods assume some xed constellation, which is not always the case. The modulation recognition procedure would require, in addition to the feature extraction, a proper front-end and decision procedure. The task of the front-end would be the channel equalization and to produce the correct sampled signal representation before feeding the measurements to the feature extraction system. The decision procedure takes the features as an input and outputs the most probable modulation type which has produced the measured signal. As the front-end depends heavily on the application, the future work will concentrate on the decision procedures.

References [1] P.O. Amblard, M. Gaeta and J.L. Lacoume, Statistics for Complex Variables and Signals - Part II: Signals, Signal Processing, vol. 53, no. 1, pp. 1525, August 1996. [2] K. Assaleh, K. Farrell and R.J. Mammone, A New Method of Modulation Classication for Digitally Modulated Signals, Proceedings of IEEE MILCOM '92, vol. 2, pp. 712716, October 1992. [3] E.E. Azzouz and A.K. Nandi, Automatic Modulation Recognition of Communication Signals, Kluwer Academic Publishers, 1996. [4] E.E. Azzouz and A.K. Nandi, Procedure for Automatic Recognition of Analogue and Digital Modulations, IEE Proceedings Communication, vol. 143, no. 5, pp. 259266, October 1996. [5] D. Boiteau and C. Le Martret, A General Maximum Likelihood Framework for Modulation Classication, Proceedings of ICASSP '98, vol. 4, pp. 2165 2168, May 1998. [6] A.B. Carlson, Communication Systems, 3rd ed., McGraw-Hill, 1986. [7] Y.T. Chan and L.G. Gadbois, Identication of the Modulation Type of a Signal, Signal Processing, vol. 16, no. 2, pp. 149154, February 1989. [8] K.M. Chugg, C-S. Long and A. Polydoros, Combined Likelihood Power Estimation and Multiple Hypothesis Modulation Classication, Proceedings of ASILOMAR-29, vol. 2, pp. 11371141, November 1995. [9] C-D. Chung and A. Polydoros, Envelope-Based Classication Schemes for Continuous-Phase Binary Frequency-Shift-Keyed Modulations, Proceedings of IEEE MILCOM '94, vol. 3, pp. 796800, October 1994. [10] I. Druckmann, E.I. Plotkin and M.N.S. Swamy, Automatic Modulation Type Recognition, IEEE Canadian Conference on Electrical and Computer Engineering, vol. 1, pp. 6568, May 1998. [11] W.A. Gardner and L.E. Franks, Characterization of Cyclostationary Random Signal Processes, IEEE Transactions on Information Theory, vol. IT-21, no. 1, pp. 414, January 1975.

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Appendix A First and Second-Order Statistics of Digital Modulated Signals In this appendix, we derive the rst and second-order statistics and cyclic autocorrelations for digital modulated signals. The modulation types under consideration are M-ary amplitude, phase, and frequency shift keyed processes. Also the carrier wave process is dealt for consistency. The statistics are derived for continuous-time processes with discrete symbol sequences. Oversampling with respect to symbol rate has to be used when the cyclic features are exploited with sampled sequences. The denition for the cyclic autocorrelation is presented in 3.1.2. In subsequent sections, we let M be the number and S[m] be an i.i.d random sequence of the symbols with uniform probability mass function (PMF). Then we can write

S[m] ∈ 0, 1, . . . , M − 1, £ ¤ 1 P S[m] = a = , M

M = 1, 2, 4, 8, . . . ,

a = 0, 1, . . . , M − 1,

(A.1)

where m ∈ Z is the index of the mth symbol.

A.1 Carrier Wave The complex envelope representation of the carrier wave (CW) process is C(t) = e0 = 1. The mean function and time-average is therefore trivially periodic µC (t) = µC (t + T ) = µC = 1 with any period T . The autocorrelation function is also trivially periodic RC (t1 , t2 ) = RC (t1 + T, t2 + T ) = RC (t + τ2 , t − τ2 ) = 1, where τ = t1 − t2 . The cyclic autocorrelation function can be expressed as

APPENDIX A. FIRST AND SECOND-ORDER STATISTICS OF DIGITAL MODULATED SIGNALS 73

RCα (τ )

1 = T

Z

∞

e−j2παt dt =

−∞

2π δ(2πα), T

(A.2)

where the last identity follows from the Fourier transform pair 1 ↔ 2πδ(ω).

A.2 Amplitude Shift Keying Let A[m] = 2S[m] + 1 − M , the complex envelope representation of the M-ary amplitude shift keying (MASK) process can be expressed as ∞ X

C(t) =

A[m]g(t − mT )

m=−∞ ∞ X £

=

¤ 2S[m] + 1 − M g(t − mT ),

(A.3)

m=−∞

where the distance between adjacent symbols is 2d = 2. The mean function of the MASK process can be expressed as ∞ X £

£ ¤ ¤ 2ES S[m] + 1 − M g(t − mT ) = 0,

(A.4)

£ ¤ 1 1 M (M − 1) M −1 ES S[m] = [0 + 1 + · · · + (M − 1)] = = , M M 2 2

(A.5)

µC (t) = ES [C(t)] =

m=−∞

because

where we have used the identity for the sum of integers 1 + 2 + 3 + · · · + n = n(n+1) . 2 Thus the mean function and time-average of the MASK process is trivially periodic µC (t) = µC (t + T ) = µC = 0. The autocorrelation function of the MASK process can be expressed as ∞ X ¤ RC (t1 , t2 ) = ES C(t1 )C(t2 ) =

£

∞ X

£ ¤ ES A[m]A[n] g(t1 − mT )g(t2 − nT )

m=−∞ n=−∞ ∞ X 1 2 = (M − 1) g(t1 − kT )g(t2 − kT ), 3 k=−∞

(A.6)

APPENDIX A. FIRST AND SECOND-ORDER STATISTICS OF DIGITAL MODULATED SIGNALS 74 £ ¤ where the expectation ES A[m]A[n] can be expanded as £ ¤ £ ¤ £ ¤ ES A[m]A[n] = ES (2S[m] + 1 − M )(2S[n] + 1 − M ) = 4ES S[m]S[n] £ ¤ £ ¤ £ ¤ £ ¤ +2ES S[m] + 2ES S[n] − 2M ES S[m] − 2M ES S[n] +M 2 − 2M + 1,

(A.7)

£ ¤ and the expectation ES S[m]S[n] for m 6= n, by Equation (A.5) and independent symbols, can be expressed as £ ¤ £ ¤ £ ¤ (M − 1)2 ES S[m]S[n] = ES S[m] ES S[n] = , 4

(A.8)

£ ¤ and therefore Equation (A.7) is zero if m 6= n. The expectation ES S[m]S[n] for m = n = k can be expressed as £ ¤ £ ¤ ¤ 1£ 2 ES S[m]S[n] = ES S 2 [k] = 0 + 12 + · · · + (M − 1)2 M =

(M − 1)(2M − 1) , 6

(A.9)

where we have used the identity for the sum of squared integers 12 + 22 + 32 + · · · + . After some calculus, Equation (A.7) becomes n2 = n(n+1)(2n+1) 6

£ ¤ 1 ES A[m]A[n] = (M 2 − 1)δ(m − n). 3

(A.10)

Thus the double summation in Equation (A.6) can be replaced by one summation over k = m = n. Let the signal pulse be a square g(t) = u(t) − u(t − mT ), where T is the symbol duration. The autocorrelation function can now be expressed as ∞ X 1 2 RC (t1 , t2 ) = (M − 1) [u(t1 − kT ) − u(t1 − (k + 1)T )][u(t2 − kT ) 3 k=−∞

½1 −u(t2 − (k + 1)T )] =

(M 2 − 1), 0, 3

¥ t1 ¦

¥ ¦ = tT2 , otherwise T

(A.11)

where b·c denotes the integer part of the argument. The autocorrelation function is invariant to a shift by T ; i.e. periodic. Thus the MASK process is wide-sense cyclostationary.

APPENDIX A. FIRST AND SECOND-ORDER STATISTICS OF DIGITAL MODULATED SIGNALS 75 By substitution τ = t1 − t2 and examination of the bounds of the products of the unit step functions, the autocorrelation function can be expressed as ∞ X ¤ £ τ τ 1 |τ | |τ | 2 u(t − − kT ) − u(t + − (k + 1)T ) RC (t + , t − ) = (M − 1) 2 2 3 2 2 k=−∞

×[u(τ + T ) − u(τ − T )],

(A.12)

where the last term of the product causes the autocorrelation to be zero when the lag is larger than T . Due to its periodicity, the autocorrelation function can be expanded to its Fourier series and the coecients can be obtained as follows

RCα (τ )

∞ Z ∞ X ¤ £ 1 |τ | |τ | 2 = (M − 1) u(t − − kT ) − u(t + − (m + 1)T ) 3T 2 2 k=−∞ −∞

×[u(τ + T ) − u(τ − T )]e−j2παt dt

(A.13)

Z ∞ h i X ¡ −j2πα( |τ | +kT ) ¢ ∞ −|τ | 1 2 −j2πα( +(k+1)T ) 2 2 e = (M − 1) −e u(t)e−j2παt dt 3T −∞ k=−∞ ×[u(τ + T ) − u(τ − T )]

(A.14)

∞ X ¡ −j2πα( |τ | +kT ) ¢¡ 1 ¢ |τ | 1 2 2 = (M − 1) e − e−j2πα(− 2 +(k+1)T ) + πδ(2πα) 3T j2πα k=−∞

×[u(τ + T ) − u(τ − T )],

(A.15)

where α is the cyclic frequency, namely α = Tn , n = 0, ±1, ±2, . . . , and n is the number of harmonic. From Equation (A.14) we get Equation (A.15) by the timeshift property of Fourier transform f (t − τ ) ↔ e−jωτ F (ω) and Equation (A.15) 1 follows from the Fourier transform of the unit step function u(t) ↔ jω + πδ(ω) where ω = 2πf .

A.3 Phase Shift Keying The phase sequence for M-ary phase shift keying (MPSK) process can be expressed as Φ[m] = 2π S[m]. The complex envelope representation of the MPSK process will M be as follows

APPENDIX A. FIRST AND SECOND-ORDER STATISTICS OF DIGITAL MODULATED SIGNALS 76 ∞ X

C(t) =

ejΦ[m] g(t − mT )

m=−∞ ∞ X

=

ejΦ[m] [u(t − mT ) − u(t − (m + 1)T )],

(A.16)

m=−∞

where the signal pulse is assumed to be a square g(t) = u(t) − u(t − T ). The mean function of C(t) can be expressed as

µC (t) = ES [C(t)] =

∞ X

ES [ejΦ[m] ][u(t − mT ) − u(t − (m + 1))] = 0, (A.17)

m=−∞

because

ES [ejΦ[m] ] = =

2π 1 2π 1 1 j0 e + ej M + · · · + ej(M −1) M M M M

¤ M 2π 2π 1£ 0 (e + ejπ ) + (ej M + ej( 2 +1) M ) + . . . = 0. M

(A.18)

Thus the mean function and time-average of the MPSK process is trivially periodic µC (t) = µC (t + T ) = µC = 0. The autocorrelation function of the MPSK process can be expressed as

RC (t1 , t2 ) = ES [C(t1 )C(t2 )] =

∞ X

∞ X

£ j £Φ[m]+Φ[n]¤ ¤ ES e

(A.19)

m=−∞ n=−∞

×[u(t1 − mT ) − u(t1 − (m + 1)T )][u(t2 − nT ) − u(t2 − (n + 1)T )], where due to the independent symbols, the expectation inside the summations for m 6= n can be expressed as

£ j £Φ[m]+Φ[n]¤ ¤ = ES [ejΦ[m] ]ES [ejΦ[n] ] = 0, ES e and for m = n = k

(A.20)

APPENDIX A. FIRST AND SECOND-ORDER STATISTICS OF DIGITAL MODULATED SIGNALS 77 £ j £Φ[m]+Φ[n]¤ ¤ 4π 4π ¤ 1 £ j0 ES e = ES [ej2Φ[k] ] = e + ej M + · · · + ej(M −1) M M ½ 1, M = 2 = . (A.21) 0, M = 4, 8, . . . When M = 4, 8, . . . , the autocorrelation function will be£ identically zero. For £ j Φ[m]+Φ[n]¤ ¤ 2PSK process, the expectation in Equation (A.21) is ES e = δ(m − n). Thus the summations over m and n can be replaced with one summation over k = m = n and Equation (A.20) becomes ∞ X

RC (t1 , t2 ) =

[u(t1 − kT ) − u(t1 − (k + 1)T )][u(t2 − kT ) − u(t2 − (k + 1)T )]

k=−∞

½ =

1, 0,

¥ t1 ¦

¥ t2 ¦ = T T , otherwise

(A.22)

where b·c denotes the integer part of the argument. Clearly RC (t1 , t2 ) = RC (t1 + T, t2 + T ) and thus the 2PSK process is wide-sense cyclostationary. The autocorrelation function of the 2PSK process is identical to the autocorrelation function of the MASK process except the constant 31 (M 2 − 1) which can be discarded due to the linearity property of the Fourier transform. Thus the cyclic autocorrelation of the 2PSK process can be expressed as

RCα (τ ) =

∞ ¢¡ 1 ¢ −|τ | 1 X ¡ −j2πα( |τ | +kT ) 2 e − e−j2πα( 2 +(k+1)T ) + πδ(2πα) T k=−∞ j2πα

(A.23)

×[u(τ + T ) − u(τ − T )], where α is the cyclic frequency, namely α = number of harmonic.

n , T

n = 0, ±1, ±2, . . . , and n is the

A.4 Frequency Shift Keying The complex envelope representation of the M-ary frequency shift keying (MFSK) process can be expressed as

APPENDIX A. FIRST AND SECOND-ORDER STATISTICS OF DIGITAL MODULATED SIGNALS 78 © C(t) = exp jω∆

Z

t

∞ X

S[m]g(τ − mT )dτ

ª

m=−∞

Z t ∞ X © ª = exp jω∆ S[m] [u(τ − mT ) − u(τ − (m + 1)T )]dτ m=−∞ ∞ X © ª = exp jω∆ S[m](t − mT )[u(t − mT ) − u(t − (m + 1)T )] m=−∞

=

∞ X

ejω∆ S[m](t−mT ) [u(t − mT ) − u(t − (m + 1)T )],

(A.24)

m=−∞

where T is the symbol duration and the pulse shape is assumed to be a square g(t) = u(t) − u(t − mT ). The expectation with respect to S[m] can be expressed as ∞ M −1 1 X X jω∆ a(t−mT ) e [u(t − mT ) − u(t − (m + 1)T )] µC (t) = ES [C(t)] = M a=0 m=−∞ M −1 ∞ h1 1 X X jω∆ a(t−kT ) i = e [u(t + T − kT ) − u(t + T − (k + 1)T )] + M M a=1 k=−∞

(A.25)

= µC (t + T ),

where k = m + 1. Equation (A.25) shows that the mean function of the MFSK process is periodic and the time-average of the MFSK process is µC = M1 . The autocorrelation function of the MFSK process can be expressed as ∞ X

RC (t1 , t2 ) = ES [C(t1 )C(t2 )] =

∞ X

ES [ejω∆ [S[m](t1 −mT )+S[n](t2 −nT )] ]

m=−∞ n=−∞

×[u(t1 − mT ) − u(t1 − (m + 1)T )][u(t2 − nT ) − u(t2 − (n + 1)T )] M −1 ∞ 1 X X jω∆ a(t1 +t2 −2kT ) = e M a=0 k=−∞

×[u(t1 − kT ) − u(t1 − (k + 1)T )][u(t2 − kT ) − u(t2 − (k + 1)T )] = RC (t1 + T, t2 + T ),

(A.26)

APPENDIX A. FIRST AND SECOND-ORDER STATISTICS OF DIGITAL MODULATED SIGNALS 79 where the last equivalence can be obtained by substitution k = m + 1. From Equations (A.25) and (A.26) we see that the MFSK process is wide-sense cyclostationary. By substitution τ = t1 − t2 and examination of the bounds of the products of the unit step functions, the autocorrelation function becomes M −1 ∞ τ τ 1 X X jω∆ 2a(t−kT ) £ |τ | RC (t + , t − ) = e u(t − − kT ) 2 2 M a=0 k=−∞ 2

−u(t +

¤ |τ | − (k + 1)T ) [u(τ + T ) − u(τ − T )], (A.27) 2

where the terms depending on t are same as with the 2PSK process except term ejω∆ 2at . By the frequency-shift property of the Fourier transform ejω0 t f (t) ↔ F (ω − ω0 ) and by the cyclic autocorrelation function of the 2PSK process, the cyclic autocorrelation function of MFSK process can be expressed as

RCα (τ )

M −1 ∞ ¢ |τ | 1 X X ¡ −j2(πα−aω∆ )( |τ | +kT ) 2 = e − e−j2(πα−aω∆ )(− 2 +(k+1)T ) T M a=0 k=−∞

×

¡

¢ 1 + πδ(2πα − 2aω∆ ) ejω∆ 2amT j2(πα − aω∆ ) (A.28)

×[u(τ + T ) − u(τ − T )], where α is the cyclic frequency, namely α = number of harmonic.

n , T

n = 0, ±1, ±2, . . . , and n is the