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Mixed Signal Detection and Parameter Estimation Based on Mixed Signal Detection and Parameter Estimation Based on

Second-Order Cyclostationary Features Second-Order Cyclostationary Features

Dong Li Wright State University

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Mixed Signal Detection and Parameter Estimationbased on Second-Order Cyclostationary Features

A thesis submitted in partial fulfillmentof the requirements for the degree of

Master of Science in Engineering

by

Dong LiB.S., Tianjin University, 2009

2015Wright State University

Wright State UniversityGRADUATE SCHOOL

December 1, 2015

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPER-VISION BY Dong Li ENTITLED Mixed Signal Detectionand ParameterEstimationbasedon Second-OrderCyclostationaryFeatures BE ACCEPTED IN PARTIAL FUL-FILLMENT OF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinEngineering.

Zhiqiang Wu, Ph.D.Thesis Director

Brian Rigling, Ph.D.Department Chair

Committee onFinal Examination

Zhiqiang Wu, Ph.D.

Kefu Xue, Ph.D.

Yan Zhuang, Ph.D.

Robert E. W. Fyffe, Ph.D.Vice President for Research andDean of the Graduate School

ABSTRACT

Li, Dong. M.S.Egr., Department of Electrical Engineering,College of Engineering & ComputerScience, Wright State University, 2015.Mixed Signal Detection and Parameter Estimation basedon Second-Order Cyclostationary Features.

Signal detection and radio frequency (RF) parameter estimation have received a lot of

attention in recent years due to the need of spectrum sensingin many military and civilian

communication applications. In most of existing work, the target signal is assumed to be

a single RF signal with no overlapping with other RF signals.However, in a spectrally

congested and spectrally contested environment, multiplesignals are often mixed together

at the signal detector with significant overlap in spectrum.Conventional frequency anal-

ysis through Fourier transform is not capable of detecting mixed signals with significant

spectral overlap. In this thesis, we first demonstrate the feasibility of using second-order

cyclostationary feature to perform mixed signal detection. We then use the cyclostationary

features to estimate the carrier frequencies of these mixedsignals. Next, we extend our

work to higher order modulation. We develop a robust algorithm to detect mixed signals

and estimate their symbol rates as well as carrier frequencies via spectral coherence func-

tion (SOF) features. Furthermore, we evaluate the detection and estimation performances

of the proposed algorithm in various channel conditions andsignal mixture scenarios. Sim-

ulation results confirm the effectiveness of the proposed schemes.

iii

List of SymbolsChapter 1

CR Cognitive RadioDSA Dynamic Spectrum AccessRF Radio FrequencySCF Spectral Correlation FunctionSOF Spectrum Coherence Function

Chapter 2

MISO Multiple Input Single OutputE· Expected ValueRα

x(τ) Cyclic Autocorrelation FunctionSαx (f) Spectral Correlation Function

Cαx (f) Spectrum Coherence Function

C(kj) k-th order cumulant

Chapter 3

TH Threshold ValueN Number of components in mixed-signalfc Carrier FrequencyRs Symbol RateB Null-to-Null bandwidth

iv

Contents

1 Chapter 1: Introduction 11.1 Signal Detection, RF Parameter Estimation and Mixed Signal Detection/RF Parameter Estimation1.2 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Thesis Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Chapter 2: Mixed Signal Detection and Second-Order Cyclostationary Theory 42.1 Signal Detection and RF Parameter Estimation. . . . . . . . . . . . . . . 42.2 Mixed Signal Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Second-order Cyclostationary Features. . . . . . . . . . . . . . . . . . . . 6

2.3.1 Cyclostationary Autocorrelation Function. . . . . . . . . . . . . . 62.3.2 Spectral Correlation Function (SCF). . . . . . . . . . . . . . . . . 72.3.3 Spectrum Coherence Function (SOF). . . . . . . . . . . . . . . . 7

2.4 Higher-order Features (Fourth-order Cumulant Features) . . . . . . . . . . 8

3 Chapter 3: Mixed-signal Detection and Carrier Frequency Estimation 93.1 Mixed-signal overlapping model. . . . . . . . . . . . . . . . . . . . . . . 93.2 Mixed-signal Detection based on SCF. . . . . . . . . . . . . . . . . . . . 93.3 Performance of SCF based Mixed Signal Detection. . . . . . . . . . . . . 13

3.3.1 Estimate the Kratio Value. . . . . . . . . . . . . . . . . . . . . . 133.3.2 Mixed Signal with Two Components. . . . . . . . . . . . . . . . . 143.3.3 Mixed Signal with Two Components in Different ChannelConditions 143.3.4 Mixed Signal with Two Components with Different Spectrum Overlap 153.3.5 Mixed Signal with More Components. . . . . . . . . . . . . . . . 153.3.6 Mixed Signal with More Components in Different Noise Conditions 17

3.4 Mixed Signal Detection based on SOF. . . . . . . . . . . . . . . . . . . . 183.5 Performance of SOF based mixed signal detection. . . . . . . . . . . . . . 20

3.5.1 Comparison Between SOF and SCF based Algorithms. . . . . . . 203.5.2 Effects of Different Channel Conditions and Noises. . . . . . . . . 213.5.3 Resistance to Different Spectrum Overlap. . . . . . . . . . . . . . 233.5.4 Mixed Signal with More Signal Components. . . . . . . . . . . . 23

v

4 Chapter 4: Mixed Signal Detection for Higher Order Modulations 254.1 SCF and SOF for Mixed Signals with Higher-order Modulations . . . . . . 254.2 Fourth-order Cumulant based method for detection. . . . . . . . . . . . . 264.3 SCF and SOF for M-QAM signal. . . . . . . . . . . . . . . . . . . . . . . 274.4 SOF based Mixed Signal Detection and Symbol Rate Estimation . . . . . . 324.5 Performance of SOF based Mixed Signal Detection and Symbol Rate Estimation34

4.5.1 Detection of Different Components with Different Modulation Types344.5.2 Effects of Different Channel Conditions and Noises. . . . . . . . . 354.5.3 Mixed Signal with More Signal Components. . . . . . . . . . . . 374.5.4 Threshold for Detection. . . . . . . . . . . . . . . . . . . . . . . 37

5 Conclusion 39

Bibliography 40

vi

List of Figures

2.1 Signal Detection Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Spectrum of mixed signal with two different components. . . . . . . . . . 5

3.1 Spectra of two signals with 100% spectrum overlap. . . . . . . . . . . . . 103.2 SCF of two different BPSK signals with 90% spectrum overlap . . . . . . . 113.3 The probability density function of Kratio. . . . . . . . . . . . . . . . . . 133.4 SCF of two different BPSK Signals with 0% and 95% spectrumoverlap . . 143.5 SCF of two different BPSK signals with 95% spectrum overlap in flat fading channel153.6 SCF of two different BPSK signals with 99% spectrum overlap in flat fading channel163.7 SCF of mixed BPSK signal with more than two components in 95% spectrum overlap163.8 SCF of three different BPSK signals with 95% spectrum overlap in AWGN channel173.9 SCF of three different BPSK signals with 95% spectrum overlap in Flat Fading channel173.10 SOF of two different BPSK signals with 100% spectrum overlap . . . . . . 203.11 SCF and SOF of two different BPSK signals with 7 dB received power difference213.12 SOF of two different BPSK signals with different received power differences223.13 Received-power differences vs. Detection accuracy for two components. . 233.14 SOF of two different BPSK signals with 7 dB received power difference . . 243.15 SOF and Detection accuracy for three components in flat fading channel. . 24

4.1 SCF of BPSK and QPSK signal. . . . . . . . . . . . . . . . . . . . . . . 264.2 C42 values for signal components with different Received Power Differences274.3 SOF of different modulated signal whenf = fc andα ∈ (0, 1.2Rs) . . . . 304.4 SOF of two different signal components with 100% spectrum overlap . . . 344.5 SOF of two signals with 0 dB received power difference. . . . . . . . . . 354.6 SOF of two different signals with different received power difference. . . . 364.7 Received-power differences vs. Detection accuracy in Flat fading channel. 374.8 Received-power differences vs. Detection accuracy fordifferent threshold value38

vii

Chapter 1: Introduction

1.1 Signal Detection, RF Parameter Estimation and Mixed

Signal Detection/RF Parameter Estimation

Signal detection and radio frequency (RF) parameter estimation play a very important role

in many military and civilian communication applications,e.g., signal identification, in-

terference identification, spectrum management, communication surveillance, electronic

countermeasures, and so on [1]. Recent development in cognitive radio (CR) and dynamic

spectrum access (DSA) network has also brought strong interest to signal detection and RF

parameter estimation of unknown RF signals. With the rapid growing of cognitive radio

network research, spectrum sensing has been investigated thoroughly in the past decade

to detect the existence of primary users and spectrum holes to allow secondary user trans-

mission. To allow a better co-existence of primary users andsecondary users, it has been

realized that more signal characteristics are needed in addition to the mere existence of the

target signals [2,3].

More sophisticated signal detection/RF parameter estimation/modulation identifica-

tion algorithms have been proposed to not only detect the existence of the target signal but

obtain important characteristics of the target signal [1,4–8]. However, in most of existing

work, the target signal is often assumed to be a single signalwithout overlap in spectrum

with other signals. This assumption is becoming more and more invalid with the recent

1

development of cognitive radio network since multiple secondary users might try to grasp

the same spectrum hole and such signals also need to be detected and analyzed to avoid

interference. Hence, it is highly desired to develop effective scheme to detect the existence

of mixed signals with multiple users signals significantly overlap in spectrum, identify how

many signals are mixed together, and estimate their RF parameters.

1.2 Motivation

In this thesis, we first demonstrate the feasibility to detect the components of mixed signals

by employing second-order cyclostationary features, namely spectral correlation function

(SCF) and spectral coherence function (SOF). We start with the simple case of a mixed

signal which is composed of multiple BPSK modulated signalsat slightly different car-

rier frequencies. The two different signals have significant overlap in frequency domain.

Therefore, it is impossible to determine how many signals are mixed together through a

conventional spectral analysis. We show that cyclostationary analysis is capable of identi-

fying the number of individual components accurately [9].

Next, we extend this work to employ a robust algorithm to detect mixed signals and

estimate their carrier frequencies via spectral coherencefunction (SOF) features [10–16].

We demonstrate the effectiveness and efficiency of the second-order cyclostationary fea-

tures under different conditions.

Furthermore, we develop a sophisticated robust algorithm to detect mixed signals and

estimate their RF parameters when higher order modulationssuch as QPSK and 16QAM

are employed. By employing SOF features, we are able to find a stable threshold in our

mixed signal detection algorithm to determine how many mixed components are present

[17].

We have thoroughly evaluated the detection performance, RFparameter estimation

performances under various channel conditions and signal mixture scenarios. Simulation

2

results confirm the effectiveness and robustness of our algorithms.

1.3 Thesis Outline

Chapter 1 describes the basic background about signal detection, RF parameter estimation,

and mixed signal detection/RF parameter estimation. Chapter 2 introduces the mixed sig-

nal detection problem and challenges, and reviews second-order cyclostationary analysis.

Specifically, second order cyclostationary features including SCF and SOF, and fourth-

order cumulants are introduced. Chapter 3 discusses our proposed second-order cyclosta-

tionary feature based mixed signal detection algorithm andcarrier frequency estimation

algorithm. Simulation results of mixed signal detection based on SCF and SOF are also

presented. Chapter 4 discusses the mixed signal detection for signals with higher order

modulations. In this chapter, we evaluate the signal detection based on fourth-order cu-

mulant and propose a new method for higher-order modulationtype. Then we evaluate

the performances in various channel conditions and signal mixture scenarios. Simulation

results at different signal to noise ratio and channel conditions validate the advantage and

reliability of the proposed method. Conclusion follows in Chapter 5.

3

Chapter 2: Mixed Signal Detection and

Second-Order Cyclostationary Theory

2.1 Signal Detection and RF Parameter Estimation

Signal detection and RF parameter estimation have been an important topic in communi-

cation, radar, navigation, and electronic warfare. Figure2.1shows a basic signal detection

model. With the rapid growth of cognitive radio and dynamic spectrum access network

technologies, spectrum sensing has received strong interest to detect the existence of pri-

mary users (PUs) and spectrum holes. It has also been realized that to ensure more sophis-

ticated cognitive radio systems, important RF parameters such as carrier frequency, symbol

rate, and modulation type need to be detected as well.

��

��

� ��

��

��

��

��

��

� ��

��

��

Figure 2.1: Signal Detection Model

4

2.2 Mixed Signal Detection

However, in most existing work, the target signal is often assumed to be a single commu-

nication signal without overlap with other signals in the frequency domain.

Figure2.2(a)shows the spectrum of two narrow-band BPSK signal components with-

out significant spectrum overlap. It is clear from spectrum analysis that there are two signal

components. Hence, it is feasible to use a filter to first distinguish the target signal out and

perform RF characteristic estimation tasks next to avoid the interferences from the spec-

trum environment.

However, in a spectrally congested and spectrally contested environment, multiple

signals may overlap significantly in frequency domain. Figure 2.2(b)shows the spectrum

of a mixed signal with two narrow-band BPSK signal components with very close carrier

frequencies so that there is a 95% spectrum overlap (the spectrum overlap is defined as the

overlapping spectrum divided by the bandwidth of the individual signal).

It is evident from Figure2.2(b) that we can no longer tell that there are two signal

components by simple frequency analysis, and we cannot use anarrow-band filter to dis-

tinguish one signal from the other. This is what we are tryingto achieve in this thesis: use

cyclostationary analysis to determine how many signal components exist and estimate their

parameters.

1 1.5 2 2.5 3

x 104

0

2

4

6

8

10

12

14

16

18

20

Spectrum Analysis of mixed signals Component 1: QPSK signal (Fc=15000, SymbolRate=2500) (Hz)Component 2: BPSK signal (Fc=25000, SymbolRate=2400) (Hz)

Spectrum Overlap: 0 %

Frequency (Hz)

Am

plitu

de

(a) 0% spectrum overlap

1 1.5 2 2.5 3

x 104

0

5

10

15

20

25

30

Spectrum Analysis of mixed signalsComponent 1: QPSK signal (Fc = 20000, SymbolRate = 2500) (Hz)Component 2: BPSK signal (Fc = 20100, SymbolRate = 2400) (Hz)


Frequency (Hz)

Am

plitu

de

(b) 100% spectrum overlap

Figure 2.2: Spectrum of mixed signal with two different components

5

2.3 Second-order Cyclostationary Features

The traditional signal analysis model is based on stationary random process. It has long

been recognized that many man-made signals, such as communication signal and radar

signal, are actually cyclostationary random processes instead of stationary random pro-

cesses. These man-made signals demonstrate significant periodicity in their first-order or

second-order statistic characteristics(average value, correlation function,etc).

The periodic features of these signals are not reflected in the conventional power spec-

trum. By employing cyclostationary analysis, e.g., using the correlation characteristics be-

tween different frequency bands, it is feasible to reveal these periodic features and perform

signal detection, RF parameter estimation, and even modulation detection [18].

2.3.1 Cyclostationary Autocorrelation Function

Assumex(t) is a cyclostationary signal, its correlation function is [10–15]

Rx(t+τ

2, t−

τ

2) = E[x(t +

τ

2)x∗(t−

τ

2)], (2.1)

Here,Rx(t+τ2, t− τ

2) is a periodic function with periodT .

Decompose this signal into Fourier Series, we have

Rx(t +τ

2, t−

τ

2) =

∑

α

Rαx(τ)e

j2παt (2.2)

where the Fourier coefficients are

Rαx(τ) = lim

T→∞

1

T

∫ T/2

−T/2

Rx(t+τ

2, t−

τ

2)e−j2παtdt (2.3)

whereα = m/T is the cyclic frequency,Rαx(τ) is the cyclic autocorrelation function.

6

2.3.2 Spectral Correlation Function (SCF)

The Fourier transformation of cyclic autocorrelation functionRαx(τ) is named Spectral Cor-

relation Function(SCF), which is presented as

Sαx (f) =

∫

∞

−∞

Rαx(τ)e

−j2πfτdτ. (2.4)

whereα is the cyclic frequency,f is the spectrum frequency.

2.3.3 Spectrum Coherence Function (SOF)

As shown in [9], the SCF based algorithm doesn’t provide a fixed threshold value for detec-

tion and a heuristic algorithm needs to be performed to find appropriate detection threshold.

Here, we use SOF which is a normalized version of SCF to solve this problem. Similar

to SCF, SOF is a two variable (frequency nf and cyclic frequencyα) function. Assumex(t)

is a cyclostationary signal, its SOF defined in [10–15] is:

Cαx (f) =

Sαx (f)

[

S0x

(

f +1

2α

)

∗

S0x

(

f −1

2α

)]1/2, (2.5)

It is shown that the SOF is upper-bounded by unity (shown as Equation2.6) and can

help to remove channel effect [11]:

|CαX(f)| 6 1. (2.6)

Additionally, SOF has an invariance property [11], which means it will not be affected

by linear time-invariant transformations. Hence, we can use SOF and find an universal

threshold for our mixed signal detection purpose.

7

2.4 Higher-order Features (Fourth-order Cumulant Fea-

tures)

The higher-order modulated signals (e.g., QPSK signals and16-QAM signals) have differ-

ent fourth-order cumulant features, which can be used for modulation detection [1].

The estimated second- and fourth-order cumulants of a complex-valued stationary

random processy(n) are defined as [19,20]:

C20 =1

N

N∑

n=1

|y(n)|2 (2.7)

C21 =1

N

N∑

n=1

y2(n) (2.8)

C40 =1

N

N∑

n=1

y4(n)− 3C220 (2.9)

C41 =1

N

N∑

n=1

y3(n)y∗(n)− 3C20C21 (2.10)

C42 =1

N

N∑

n=1

|y(n)|4 − |C20|2 − 2C2

21 (2.11)

whereE[y(n)] = 0.

In [1], we used the normalized fourth-order cumulant defined in Equation 2.12 to

distinguish different modulation types:

C4k =C4k

C221

, k = 0, 1, 2 (2.12)

The theoretical normalizedC42 for QPSK and 16QAM signals are [19]:

C42(QPSK) = −1.0000

C42(16QAM) = −0.6047

8

Chapter 3: Mixed-signal Detection and

Carrier Frequency Estimation

3.1 Mixed-signal overlapping model

Figure3.1shows an example of mixed signal detection with significant overlap. It is clear

from Figure3.1 that the second signal’s spectrum is entirely contained within the first

signal’s spectrum with a 100% overlap. This is the worst casescenario for spectrum over-

lapping. Conventional spectrum analysis is incapable of detecting that there are two signals

in the spectrum.

3.2 Mixed-signal Detection based on SCF

Based on second-order cyclostationary theory, we can derive the SCF of such a mixed

signal. Assume a mixed signal is composed by two BPSK signalswith slightly different

carrier frequencies:

x(t) = a1(t) cos(2πf1t + φ1) + a2(t) cos(2πf2t+ φ2) (3.1)

wherea1(t) anda2(t) are the envelope of the two individual signals,f1 andf2 are the

9

��

� ��

� ��

��

�� !�"��#��

Figure 3.1: Spectra of two signals with 100% spectrum overlap

carrier frequencies which are very close, andφ1 andφ2 are the phases of the two signals.

Equation3.2 shows that whenα = ±2fc1 andα = ±2fc2, Sαa (f) has maximum

value [9].

Sαx (f) =

1

4

[

Sαa1(f + f1) + Sα

a1(f − f1) + Sα

a2(f + f2) + Sα

a2(f − f2)

]

, α = 0

1

4Sα−2f1a1

(f)e+j2φ1, α = +2f1

1

4Sα+2f1a1 (f)e−j2φ1, α = −2f1

1

4Sα−2f2a2 (f)e+j2φ2, α = +2f2

1

4Sα+2f2a2

(f)e+j2φ2, α = −2f2

0, otherwise

(3.2)

According to this feature, we can now easily identify how many signals are co-existing

in the mixed signal. The following low complexity algorithmsummarizes the procedure of

our mixed signal detection:

10

1. Based on spectrum analysis of signals, we can coarsely estimate the mixed signal’s

carrier frequencyfc and bandwidth.

2. Next, we can set an appropriate scope ofα values in order to reduce the amount of

computation. However, in order to obtain the accurate valueof SCF, the resolution

of α should be chosen at its highest.

3. We then calculate the SCF of the mixed signal.

4. The number of signals and the carrier frequencies can thenbe estimated via second-

order cyclostationary features.

−2Fc2 −2Fc10

0.5

1

1.5

2

2.5x 10

6

α

SC

F

Spectral correlation magnitude features for mixed signals

2 BPSK signals with Spectrum Overlap: 90 %

alpha=−16220 Hz

alpha=−8220 Hz

P1

P2

win 2

win 1

win 2

HH

Figure 3.2: SCF of two different BPSK signals with 90% spectrum overlap

Figure3.2shows the SCF of a mixed signal. An algebraic algorithms for identifying

how many different signals are in the mixed signal has been proposed as following:

1. Find the extrema of the maximum peaks of the SCF.

2. Find a maximum valueP1 of the peaks in ’win1’ area.

11

3. Due to the influence of the peak side-lobes, we give up part of values which are very

close to the maximum peak. ValueH, the length of this abandoned area, should be

much less then the frequency difference between two adjacent signal in the actual

situation.

4. Find a maximum valueP2 in the rest area ’win2’.

5. Compare P1 and P2, and define a ratio value ’Kratio’ using Equation3.3

Kratio =P2

P1(3.3)

where P1 is the maximum peak, P2 is the secondary maximum peak.

6. According to lots of experiments estimate the reliable Kratio.

7. Find the threshold valueTH

TH = P1×Kratio (3.4)

8. Make a decision of the signal numberN of the mixed signal using a threshold value

TH

N =

2, if two values ofSCF > TH

1, if one values ofSCF > TH(3.5)

If the mixed signal has more than two components, we can use the same algebraic

algorithm to calculate the number of signals:

1. Find the maximum peak P1.

2. Find the threshold TH using Equation3.4

12

3. Make decision using threshold value TH

Number of signals= N

whereN is the number of peaks whose value is larger than TH.

3.3 Performance of SCF based Mixed Signal Detection

3.3.1 Estimate the Kratio Value

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

2

4

6

8

10

12The probability density function of kratio for two signals and one signal

kratio value

PD

F

2 signals (20 dB 1:0.5)1 signal (20 dB)

K

Figure 3.3: The probability density function of Kratio

After thorough experiments, we obtain the probability density function (pdf) of Kratio

for two signals with 90% spectrum overlap and one signal, as shown in Figure3.3. For the

mixed signal, here we assume that it is in flat fading channel.The fading factors of the two

components are 1 and 0.5, and the SNR of each signal’s channelis 20 dB. For the single

signal, we assume that its fading factor is 1 and the SNR is also 20 dB. The resolution of

α is 0.05 Hz. It is clear from Figure3.3 that the Kratio value should be set toK = 0.27.

After we find the maximum peak P1, we then use K to calculate thethreshold value TH. If

other extrema peaks are larger than TH, those peaks should represent signal components.

13

3.3.2 Mixed Signal with Two Components

In Figure3.4(a), the signal is composed of two different BPSK signals with 0%spectrum

overlap. In comparison, Figure3.4(b)shows the SCF of a mixed signal, which is composed

of two BPSK signals with different carrier frequencies with95% spectrum overlap. It is

evident that we can find that there are two peaks atα = −2fc1 andα = −2fc2 .

−2Fc2 −2Fc10

1

2

3

4

5

6x 10

6

α (Hz)

SC

F



alpha=−80000 Hz alpha=−40000 Hz

(a) 0% spectrum overlap

−2Fc2 −2Fc10

1

2

3

4

5

6x 10

6

α (Hz)

SC

F



alpha=−42000 Hz

alpha=−40000 Hz

(b) 95% spectrum overlap

Figure 3.4: SCF of two different BPSK Signals with 0% and 95% spectrum overlap

3.3.3 Mixed Signal with Two Components in Different Channel Con-

ditions

Figure3.5shows the SCF when the mixed signals are under fading. The fading factor for

component 1 isα1 = 1, and for component 2 isα2 = 0.5. The SNR for each component in

the mixed signals are 20 dB (in Figure3.5(a)) and 10 dB (in Figure3.5(b)).

From these figures, it is evident that the proposed SCF based mixed signal detection

algorithm is robust to channel conditions and noises.

14

−2Fc2 −2Fc10

1

2

3

4

5

6x 10

6

α (Hz)

SC

FSpectral correlation magnitude features for mixed signals

(In Flat Fading Channe: SNR = 20 dB) 2 BPSK signals with Spectrum Overlap: 95 %

alpha=−42000 Hz

alpha=−40000 Hz

(a) SNR = 20 dB

−2Fc2 −2Fc10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

6

α (Hz)

SC

F

Spectral correlation magnitude features for mixed signals (In Flat Fading Channe: SNR = 10 dB)


alpha=−42000 Hz

alpha=−40000 Hz

(b) SNR = 10 dB

Figure 3.5: SCF of two different BPSK signals with 95% spectrum overlap in flat fadingchannel

3.3.4 Mixed Signal with Two Components with Different Spectrum

Overlap

Figure3.6shows the SCF of the scenario when two components of the mixedsignal have

larger spectrum overlap (99% overlap). The signal is transmitted in a flat fading channel.

The fading factor for component 1 isα1 = 1, and for component 2 isα2 = 0.5. The SNR

for each component in the mixed signal is 20 dB. It is evident that the SCF features are also

robust to the the overlap between the two spectrum components.

3.3.5 Mixed Signal with More Components

Now let’s evaluate the scenario when more than two signals are mixed together with sig-

nificant spectrum overlap.

Figure3.7(a)shows the SCF features for mixed signal with three spectrum compo-

nents. We can clearly observe the three peaks in the SCF.

Figure3.7(b)shows the SCF features for mixed signal with five components where

the spectrum overlap between them is 95%. Again, 5 peaks can be easily spotted from the

SCF.

15

−2Fc2 −2Fc10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

6

α (Hz)

SC

F



alpha=−40400 Hz

alpha=−40000 Hz

Figure 3.6: SCF of two different BPSK signals with 99% spectrum overlap in flat fadingchannel

−2Fc3 −2Fc2 −2Fc10

1

2

3

4

5

6x 10

6

α (Hz)

SC

F



alpha=−44000 Hz

alpha=−42000 Hz

alpha=−40000 Hz

(a) Mixed signal with 3 components

−2Fc5 −2Fc4 −2Fc3 −2Fc2 −2Fc10

1

2

3

4

5

6

7x 10

6

α (Hz)

SC

F



alpha=−48000 Hz

alpha=−46000 Hz

alpha=−44000 Hz

alpha=−42000 Hz

alpha=−40000 Hz

(b) Mixed signal with 5 components

Figure 3.7: SCF of mixed BPSK signal with more than two components in 95% spectrumoverlap

16

−2Fc3 −2Fc2 −2Fc10

1

2

3

4

5

6x 10

6

α (Hz)

SC


(In AWGN Channe: SNR = 20 dB) 3 BPSK signals with Spectrum Overlap: 95 %

alpha=−44000 Hz

alpha=−42000 Hz

alpha=−40000 Hz

(a) SNR = 20 dB

−2Fc3 −2Fc2 −2Fc10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

6

α (Hz)

SC

F

Spectral correlation magnitude features for mixed signals (In AWGN Channe: SNR = 10 dB)


alpha=−44000 Hz alpha=−42000 Hz

alpha=−40000 Hz

(b) SNR = 10 dB

Figure 3.8: SCF of three different BPSK signals with 95% spectrum overlap in AWGNchannel

3.3.6 Mixed Signal with More Components in Different Noise Condi-

tions

We now evaluate the performance of our proposed algorithm under different noise con-

ditions. Figure3.8 shows the case where the signals are transmitted in AWGN channels.

Each component in the mixed signals has the same SNR, which is20 dB (in Figure3.8(a))

and 10 dB (in Figure3.8(b)). We can see that in AWGN channel, the noises have little

effects on the SCF features.

−2Fc3 −2Fc2 −2Fc10

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

6

α (Hz)

SC

F



alpha=−44000 Hz

alpha=−42000 Hz

alpha=−40000 Hz

(a) SNR = 20 dB

−2Fc3 −2Fc2 −2Fc10

1

2

3

4

5

6x 10

6

α (Hz)

SC

F



alpha=−44000 Hz

alpha=−42000 Hz

alpha=−40000 Hz

(b) SNR = 10 dB

Figure 3.9: SCF of three different BPSK signals with 95% spectrum overlap in Flat Fadingchannel

17

Figure3.9 shows the case where the signals are transmitted in flat fading channels.

The fading factors for different signal components are different, which areα1 : α2 : α3 =

1 : 0.8 : 0.6. Each component in the mixed signal has the same SNR, which is20 dB (in

Figure3.9(a)) and 10 dB (in Figure3.9(b)). Under these conditions, we find out that when

the fading factor in flat fading channel is reducing to 0.6, the SCF features become less

obvious.

3.4 Mixed Signal Detection based on SOF

So far we have demonstrated the feasibility of detecting mixed signals with significant

spectrum overlap using second order cyclostationary spectrum correlation function (SCF)

features. However, the SCF based algorithm doesn’t providea fixed threshold value for

detection and a heuristic algorithm needs to be performed tofind appropriate detection

threshold.

Here, we use cyclostationary spectrum coherence function (SOF) which is a normal-

ized version of SCF to solve this problem. Assumex(t) is a cyclostationary signal, its SOF

is defined in Equation2.5.

We have known that the SOF is upper-bounded by unity and can help to remove chan-

nel effect [11].

Here, we just focus onf = 0 projection of the SOF because for typical communica-

tion signals, the SOF exhibits special features in the cyclic frequency domain whenf = 0.

Assume the received mixed signal is composed by two independent BPSK signals with

carrier frequenciesfc1 andfc2, which are different but very close to each other. Also, we

assume that they have different but close symbol rates. The two signals have indepen-

dent amplitudes, phases and initial time delays, and they transmit through two independent

fading channels.

After normalization ofSαa (f) using Equation2.5, we obtain the SOFCα

a (f). Next, we

18

use the following three-step algorithm to detect signals and estimate carrier frequencies.

1. Estimate the mixed signal’s carrier frequencyfc and bandwidthB through spectrum

analysis.

2. Calculate the SOF around2fc for a cyclic frequency span ofB with fine resolution.

3. Identify the number of signals and estimate the carrier frequencies.

The first step is a simple spectrum analysis and the carrier frequency estimationfc is

not very accurate. This estimation is only used for the second step to perform high resolu-

tion SOF calculation. Although SOF calculation is a computational demanding task with

high resolution, we are not obtaining the entire SOF across all cyclic frequency domain.

Instead, we are only obtaining SOF at those cyclic frequencies where the SOF features at

twice the carrier frequency of each and every individual signal component are expected to

exhibit. By doing this, we can significantly reduce the computational complexity.

Figure3.10shows the SOF of a mixed signal with two BPSK modulated components

with 100% overlap in spectrum. The carrier frequencies are set to fc1 = 20000 Hz and

fc2 = 20100 Hz. The symbol rates are set toRs1 = 2500 Hz andRs2 = 2400 Hz. A raised

cosine filter with roll-off factor of 1 is used for both transmitted signals. Here we assume

the power of the two components in the received signal are 3 dBdifferent from each other.

It is evident from Figure3.10 that the two signal components exhibit two peaks at

twice their carrier frequencies as predicted by the cyclostationary analysis theory. It is

also clear that the SOF peaks are way above the noise floor, hence we can set a universal

threshold to determine the number of signal components present in the target. Here, we

simply choose a predetermined threshold of0.2 and count the number of peaks above this

threshold, then estimate the carrier frequency from the cyclic frequencies of such peaks.

19

2Fc1 2Fc2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

α (Hz)

SO

F

Spectral coherence features for mixed signals (In Flat Fading Channe: SNR = −10 dB) 2 signals with Spectrum Overlap: 100 %

Signal : BPSK Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0Signal : BPSK Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.24565

alpha=40000 Hz

alpha=40200 Hz

TH = 0.2

P2

P1

Figure 3.10: SOF of two different BPSK signals with 100% spectrum overlap

3.5 Performance of SOF based mixed signal detection

3.5.1 Comparison Between SOF and SCF based Algorithms

Let’s first compare the algorithm based on SOF with the previously proposed SCF based

detection algorithm. Figure3.11show the SCF and SOF for the same mixed signal which

has two BPSK signal components. The two components have 81.25% spectrum overlap

and the power of the second one is 7 dB less than that of the firstone. The SNR of mixed-

signal is set to be 10 dB (here we define SNR as the ratio betweentotal power of received

signal and noise power).

It is evident from the figures that the SOF has a better distinction than SCF and is

more robust, in addition to the benefit of universal detection threshold.

20

2Fc1 2Fc2

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 106

α (Hz)

SC


(In Flat Fading Channe: SNR = 10 dB) 2 signals with Spectrum Overlap: 81.25 %

Signal : BPSK Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0Signal : BPSK Fc2=21000 Fb2=2400 tau2=0.1 phy2=2*pi*0.55461

alpha=40000 Hz

alpha=42000 Hz

(a) SCF

2Fc1 2Fc2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

α (Hz)

SO

F

Spectral coherence features for mixed signals (In Flat Fading Channe: SNR = 10 dB)

2 signals with Spectrum Overlap: 81.25 %Signal : BPSK Power = 1 Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0

Signal : BPSK Power = 0.19953 Fc2=21000 Fb2=2400 tau2=0.1 phy2=2*pi*0.55461

alpha=40000 Hz

alpha=42000 Hz

(b) SOF

Figure 3.11: SCF and SOF of two different BPSK signals with 7 dB received power differ-ence

3.5.2 Effects of Different Channel Conditions and Noises

The most important factor in mixed signal detection in SOF based method is the difference

of received power between different signal components. In realistic channels, the transmit-

ted power of signals, the distances between transmitters and receiver, and the fading factor

of the channels are quite different. As a direct result, at the receiver side, the received

signals’ power may be significantly different from others and the weak signal component

might be overshadowed by stronger ones. Here, we evaluate the performance under differ-

ent channel conditions.

Figure 3.12 shows the SOF of two BPSK signal components with 100% spectrum

overlap. The SNR of them are set to be 10 dB. All the parametersfor these experiments

are set to be the same except the relative power of the two components. In Fig.3.12(a), the

received power of the two components are the same; in Figure3.12(b), the difference of

received power between the two components is 5 dB; and in Figure3.12(c), the difference is

10 dB. Clearly, we can find that as the increasing of the received power difference between

two components, the detection becomes more difficult. In Figure 3.12(c), the detection

algorithm fails to detect the weak signal component.

Hence, let’s investigate the detection accuracy as a function of the relative power be-

21

2Fc1 2Fc2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

α (Hz)

SO

FSpectral coherence features for mixed signals

(In Flat Fading Channe: SNR = 10 dB) 2 signals with Spectrum Overlap: 100 %

Signal : BPSK Power = 1 Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0Signal : BPSK Power = 1 Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.77622

alpha=40000 Hz alpha=40200 Hz

(a) 0 dB received power difference

2Fc1 2Fc2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

α (Hz)

SO

F

Spectral coherence features for mixed signals (In Flat Fading Channe: SNR = 10 dB) 2 signals with Spectrum Overlap: 100 %

Signal : BPSK Power = 1 Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0Signal : BPSK Power = 0.31623 Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.68806

alpha=40000 Hz

alpha=40200 Hz

(b) 5 dB received power difference

2Fc1 2Fc2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

α (Hz)

SO

F

Spectral coherence features for mixed signals (In Flat Fading Channe: SNR = 10 dB) 2 signals with Spectrum Overlap: 100 %


alpha=40000 Hz

alpha=40200 Hz

(c) 10 dB received power difference

Figure 3.12: SOF of two different BPSK signals with different received power differences

tween the mixed signal components. Figure3.13(a)demonstrates the detection accuracy

versus the received signal power differences. Both signalsare transmitted through indepen-

dent fading channels with 10 dB received SNR. If the algorithm correctly detects the two

signals and accurately estimates their carrier frequencies, it is counted as a correct detec-

tion, otherwise it is deemed incorrect. As can be seen from the figure, when the received

power difference is less than 6 dB, our method has excellent accuracy which is higher than

90%. As the difference increases to 10 dB, the detection accuracy gradually falls to around

55%.

Figure3.13(b)shows the detection/estimation result when SNR is 0 dB. Compared

with Figure3.13(a), we find that the SNR of channel has little effect on the performance.

This is the direct result of the noise resistance property ofcyclostationary analysis, with

22

0 2 4 6 8 100

10

20

30

40

50

60

70

80

90

100

Received−power differences between two components (dB)

Det

ectio

n ac

cura

cy (

%)

Received−power differences vs. Detection accuracyfor two components in Flat fading channel (SNR = 10dB)

(a) SNR = 10 dB

0 2 4 6 8 100

10

20

30

40

50

60

70

80

90

100


Det

ectio

n ac

cura

cy (

%)

Received−power differences vs. Detection accuracyfor two components in Flat fading channel (SNR = 0dB)

(b) SNR = 0 dB

Figure 3.13: Received-power differences vs. Detection accuracy for two components

noise having no spectrum correlation.

3.5.3 Resistance to Different Spectrum Overlap

Figure 3.14 illustrates the SOF of the mixed signal with larger spectrumoverlap up to

100%. The received power difference is set as 7 dB apart and the SNR is set as 20 dB.

Compared with the case of 81.25% shown in Figure3.11(b), it is observed that the perfor-

mances are almost the same. This indicates that the proposedmixed signal detection and

carrier frequency estimation algorithm is robust to significant spectrum overlap.

3.5.4 Mixed Signal with More Signal Components

Figure3.15(a)shows SOF in the case of three BPSK modulated signal components in the

mixed signal and every two adjacent components have 100% spectrum overlap. As can be

seen from the figure, all three components are detected correctly.

Fig. 3.15(b)shows the detection performance for the case of 3 mixed components. It

is observed that when the largest received-power differences between each two components

is larger than 3 dB, the detection accuracy will drop to below90%.

23

2Fc1 2Fc2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

α (Hz)

SO

FSpectral coherence features for mixed signals

(In Flat Fading Channe: SNR = 10 dB) 2 signals with Spectrum Overlap: 100 %


alpha=40000 Hz

alpha=40200 Hz

Figure 3.14: SOF of two different BPSK signals with 7 dB received power difference

2Fc1 2Fc2 2Fc3

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

α (Hz)

SO

F

SOF features for mixed signals (In Flat Fading Channe: SNR = 10 dB) 3 signals with Spectrum Overlap: 100 %

alpha=40000 Hz

alpha=40200 Hz

alpha=40400 Hz

(a) SOF of three different BPSK signals with 100%spectrum overlap

0 1 2 3 4 5 60

10

20

30

40

50

60

70

80

90

100

largest Received−power differences among 3 components (dB)

Det

ectio

n ac

cura

cy (

%)

Received−power differences vs. Detection accuracyfor 3 components in Flat fading channel (SNR = 5dB)

(b) largest Received-power differences vs. Detectionaccuracy for 3 components (SNR = 5dB)

Figure 3.15: SOF and Detection accuracy for three components in flat fading channel

24

Chapter 4: Mixed Signal Detection for

Higher Order Modulations

In previous chapters, our mixed signal detection algorithms were performed on mixed sig-

nals with only BPSK components. Now let’s expand our algorithm to include higher mod-

ulation types, especially common modulation types like QPSK and 16QAM.

4.1 SCF and SOF for Mixed Signals with Higher-order

Modulations

It is well known that higher order modulations such as QPSK and QAM do not demonstrate

obvious peaks atα = 2fc in the cyclostationary domain(shown in Figure4.1) [14]. As a

direct result, we can no longer use previous method to detectthe number of components

when one or more of the signal components employ higher modulations. We need to find

other methods to detect the mixed signal.

25

Figure 4.1: SCF of BPSK and QPSK signal

4.2 Fourth-order Cumulant based method for detection

First of all, we attempt to use the hierarchical modulation detection based on fourth-order

cumulant. According to previous work, fourth-order cumulant C42 can be used to classify

the modulation between QPSK and 16-QAM [1] for single signal situation.

However, for the mixed signal scenario in more complex channel situation, the fourth-

order cumulant method does not work.

Assume our mixed signal is composed of two signal components. Figure4.2 is the

simulation result which shows theC42 value of different combinations. The x-axis shows

the changing of received power differences between two components.

As can be seen from Figure4.2, we can not decide a fixed threshold value to dis-

tinguish different curves. The conclusion can be drawn is that the fourth-order cumulant

method does not work for mixed signal detection.

26

0 2 4 6 8 10−1

−0.5

0

0.5

1

1.5

Received Power Differences (in dB)

C42

val

ues

C42 values for different combination of signal components in different Received Power Differences

BPSK & BPSKBPSK & QPSKBPSK & 16QAMQPSK & QPSKQPSK & 16QAM16QAM & 16QAM

Figure 4.2: C42 values for signal components with differentReceived Power Differences

4.3 SCF and SOF for M-QAM signal

According to Figure4.1, we noticed that both BPSK and QPSK signal have a highest

side-lobe atf = fc projection, so we are trying to use this feature to detect number of

components in the mixed signal.

Because QPSK signal can be thought as a 4-QAM signal, we can put our focus on

mixed M-QAM signal’s detection.

It is well known that any time-seriesx(t) can be expressed in the quadrature-amplitude

modulation(QAM) form

x(t) = c(t) cos(2πfct)− s(t) sin(2πfct). (4.1)

The second-order features of QAM signals are given in [11]. The cyclic autocorrela-

27

tion function ofx(t) is:

Rαx(τ) =

1

2[Rα

c (τ) +Rαs (τ)] cos(2πfcτ) +

1

2[Rα

sc(τ)− Rαcs(τ)] sin(2πfcτ)

+1

4

∑

n=−1,1

{

[Rα+2nfcc (τ)− Rα+2nfc

s (τ)] + ni[Rα+2nfcsc (τ) +Rα+2nfc

cs (τ)]}

,(4.2)

and the SCF ofx(t) is:

Sαx (f) =

1

4

∑

n=−1,1

{[Sαc (f + nfc) + Sα

s (f + nfc)] + ni[Sαsc(f + nfc)− Sα

cs(f + nfc)]}

+1

4

∑

n=−1,1

{

[Sα+2nfcc (f)− Sα+2nfc

s (f)] + ni[Sα+2nfcsc (f) + Sα+2nfc

cs (f)]}

.

(4.3)

Here,c(t) ands(t) are cyclostationary. For the special case in which the in-phase and

quadrature componentsc(t) ands(t) are joint purely stationary,x(t) is purely cyclostation-

ary with periodT0 = 1/2fc, and we have:

Sx(f) =1

4[Sc(f + fc) + Sc(f − fc) + Ss(f + fc) + Ss(f − fc)]

−1

2[Scs(f + fc)i − Ssc(f − fc)i]

(4.4)

and

Sαx (f) =

1

4[Sc(f)− Ss(f)]±

1

2iScs(f)r, α = ±2fc

with Sαx = 0 for α 6= 0 andα 6= 2fc.

In digital QAM modulation,c(t) ands(t) are PAM signals [11], which are

c(t) =

∞∑

−∞

cnq(t− nT0 − t0),

s(t) =

∞∑

−∞

snq(t− nT0 − t0),

28

For c(t), we have:

Sαc (f) =

1

T0Q(f +

α

2)Scn(f +

α

2)Q(f −

α

2)∗, α = k/T0

0, α 6= k/T0

(4.5)

in which

Q(f) =sin(πfT0)

πf,

For bits sequencecn, we have

Sαcn(f) =

Rcn(0), α = k/T0

0, α 6= k/T0

, |k| = 0, 1, 2, ...

here, we assumeRcn(0) = 1, andk is an integer.

According to the spectral correlation function, if we focusonf = fc projection of the

SCF, we can derive that:

Sαx (fc) =

T0

Msinc2(

k

2), α = k/T0, |k| = 0, 1, 2, ... (4.6)

where

sinc(x) =sin(πx)

πx.

Hence,Sαx (fc) shows a fixed feature atf = fc andα = k/T0 (shown in Figure4.1),

which can be used for our purpose of mixed signal component detection.

According to Equation2.5, we have:

Cαx (fc) = 1, whenα = k/T0. (4.7)

The SOF for BPSK, QPSK, and 16-QAM are shown in Figure4.3. Here, we set the

29

carrier frequencyfc = 20000 Hz, the null-to-null bandwidth of the signalB = 5000 Hz,

which means the symbol rateRs = 2500 Hz. The time delayt0 and phase offsetφ0 are set

to be 0. Here, we just focus onf = fc andα ∈ (0, 1.2Rs) projection.

0 500 1000 1500 2000 2500 30000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

alpha(Hz)

f(H

z)

SOF for signalSignal : BPSK Fc=20kHz, SymbolRate=2.5kHz

(a) BPSK

0 500 1000 1500 2000 2500 30000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

alpha(Hz)

f(H

z)

SOF for signalSignal : QPSK Fc=20kHz, SymbolRate=2.5kHz

(b) QPSK

0 500 1000 1500 2000 2500 30000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

alpha(Hz)

f(H

z)

SOF for signalSignal : 16QAM Fc=20kHz, SymbolRate=2.5kHz

(c) 16QAM

Figure 4.3: SOF of different modulated signal whenf = fc andα ∈ (0, 1.2Rs)

1. SOF of BPSK signal

For a BPSK signal

x(t) = a(t) cos(2πfct+ φ0),

in which

a(t) =∞∑

−∞

anq(t− nT0 − t0),

30

we have

Sαx (fc)(BPSK) =

T0

2sinc2(

k

2), α = k/T0, , |k| = 0, 1, 2, ... (4.8)

The SOF of a BPSK signal are shown in Figure4.3(a).

2. SOF of QPSK signal For QPSK modulation, we have

c(t) =

∞∑

−∞

cnq(t− nT0 − t0),

s(t) =

∞∑

−∞

snq(t− nT0 − t0),

Similarly, if we focus onf = fc projection of the SCF and SOF, we can derive out

that for QPSK we have

Sαx (fc)(QPSK) =

T0

4sinc2(

k

2), α = k/T0. (4.9)

The SOF of QPSK signal is shown in Figure4.3(b). Similarly, we set the carrier

frequencyfc = 20000 Hz, the symbol rate of QPSKRs = 2500 Hz.

3. SOF of 16QAM signal The SOF (f = fc) of 16-QAM signal is shown in Figure

4.3(c)

31

4.4 SOF based Mixed Signal Detection and Symbol Rate

Estimation

Based on second-order cyclostationary theory, we have derived the SCF of a mixed signal

consisted of multiple independent spectrum components significantly overlapping in the

frequency domain [9].

Assume the received mixed signalx(t) is composed by two independent signalsx1(t)

andx2(t) with carrier frequenciesfc1 andfc2, which are different but very close to each

other. Also, we assume that they have different but close symbol rates. The two signals

have independent amplitudes, phases and initial time delays, and they transmit through two

independent fading channels.

x(t) = x1(t) + x2(t) + n(t).

Given the independence of the frequency components in the mixed signal, the SCF

corresponds to:

Sαx (f) = Sα

x1(f) + Sα

x2(f). (4.10)

If we focus onf = fc projection, we will find that whenα = kRs(i), whereRs(i) =

1/T0(i) is the symbol rate ofi-th component,Sαx (fc) exhibits an impulse which can be

exploited to detect the existence of multiple components ofeach and every component.

After normalization ofSαx (f) using Equation2.5, we can obtain the SOFCα

x (f).

At the same time, because SCF (or SOF) is a two variable (f andα) function, and

the carrier frequencies of the two components are different, we just focus on the band

f ∈ (fc − B/2, fc + B/2) to detect the multiple components.

Next, we use the following three-step algorithm to detect signals and estimate carrier

frequencies.

32

1. Step 1: Estimate the mixed signal’s carrier frequencyfc and null-to-null bandwidth

B through spectrum analysis.

2. Step 2: Calculate the SOF aroundB/2 for a cyclic frequency span ofB with fine

resolution.

3. Step 3: Identify the number of signals and estimate the symbol rate.

The first step is a simple spectrum analysis and the carrier frequency estimationfc

is not very accurate. Although SOF calculation is a computational demanding task with

high resolution, we are not obtaining the entire SOF across all cyclic frequency domain.

Instead, we are only obtaining SOF at those cyclic frequencies where the SOF features at

the symbol rate of each and every individual signal component are expected to exhibit. By

doing this, we can significantly reduce the computational complexity.

Figure4.4(a)shows the SOF of a mixed signal with one BPSK modulated component

and one QPSK component with 100% overlap in spectrum. The carrier frequencies are set

to fc1 = 20000 Hz andfc2 = 20100 Hz. The null-to-null bandwidth are set toB1 = 5000

Hz andB2 = 4800 Hz, which means the symbol rates areRs1 = 2500 Hz andRs2 = 2400

Hz. A raised cosine filter with roll-off factor of 1 is used forboth transmitted signals. Here

we assume the power of the two components in the received signal are 3 dB different from

each other.

Figure4.4(b)is the view of Figure4.4(a)from α axis. It is evident from Figure4.4(b)

that the two signal components exhibit peaks atk times of their symbol rates as predicted

by the cyclostationary analysis theory. It is also clear that the SOF peaks are way above

the noise floor, hence we can set a universal threshold to determine the number of signal

components present in the target. Here, we simply choose a predetermined threshold of0.2

and count the number of peaks above this threshold, then estimate the symbol rates from

the cyclic frequencies of such peaks.

As we can see from Figure4.4(b), there are 2 peaks at cyclic frequencyα = 2400, 2500

33

(a) 3-D view

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X: 2400Y: 0.3652

alpha(Hz)

f(H

z)

Spectral coherence features for mixed signals (In Flat Fading Channel: SNR = 10 dB, power difference = 3 dB)

Signal 1: BPSK Fc(1)=20kHz, R

s(1)=2.5kHz, t

0(1)=0, φ

0(1)=0

Singal 2: QPSK Fc(2)=20.1kHz, R

s(2)=2.4kHz, t

0(2)=0.1,φ

0(2)=2π*0.9056


X: 2500Y: 0.6213X: 2500Y: 0.6213

TH=0.2

(b) α view

Figure 4.4: SOF of two different signal components with 100%spectrum overlap

Hz, according to Equation4.7, hence there are 2 components in this mixed signal, and their

symbol rates are2400 Hz and2500 Hz.

4.5 Performance of SOF based Mixed Signal Detection and

Symbol Rate Estimation

In this section, we evaluate the performance of the proposedmixed signal detection and

symbol rate estimation algorithm under various channel conditions, signal to noise ratios,

and different signal mixture scenarios.

4.5.1 Detection of Different Components with Different Modulation

Types

The most significant improvement of this method compare withthat proposed in our pre-

vious work is that, we extend our component’s number detection from only BPSK mod-

ulation to more modulation types (M-QAM). Figure4.5(a)shows the SOF of two QPSK

34

signal components and Figure4.5(b)shows that of two 16QAM components. As we can

see from the figures, our method can detect different components with different modulation

types.

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X: 2500Y: 0.5233

alpha(Hz)

f(H

z)

Spectral coherence features for mixed signals (In AWGN Channel: SNR = 10 dB)

Signal 1: QPSK Fc1

=20kHz, Rs1

=2.5kHz, t01

=0, φ01

=0Singal 2: QPSK F

c2=20.1kHz, R

s2=2.4kHz, t

02=0.1,φ

02=1.4π


X: 2400Y: 0.502

TH=0.2

(a) two QPSK signals

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

alpha(Hz)

f(H

z)


Signal 1: 16QAM Fc1

=20kHz, Rs1

=2.5kHz, t01

=0, φ01

=0Singal 2: 16QAM F

c2=20.1kHz, R

s2=1.2kHz, t

02=0.1,φ

02=2π*0.7061


TH=0.2

(b) two 16-QAM signals

Figure 4.5: SOF of two signals with 0 dB received power difference

4.5.2 Effects of Different Channel Conditions and Noises

The most important factor in mixed-signal detection in thismethod is the difference of

received power between different signal components. In realistic environment, the trans-

mitted power of signals, the distances between transmitters and receiver, and the fading

factor of the channels are quite different. As a direct result, at the receiver side, the re-

ceived signals’ power may be significantly different from others and the weak signal com-

ponent might be overshadowed by stronger ones. Here, we evaluate the performance under

different channel conditions.

Figure4.6 shows the SOF of two signal components with 100% spectrum overlap.

The SNR of them are set to be 10 dB. All the parameters for theseexperiments are set

to be the same except the relative power of the two components. In Figure4.6(a), the

difference of received power between the two components is 5dB; and in Figure4.6(b),

35

the difference is 10 dB. Clearly, we can find that as the increasing of the received power

difference between two components, the detection becomes more difficult. In Figure4.6(b),

the detection algorithm fails to detect the weak signal component.

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

alpha(Hz)


Signal 1: BPSK Fc1

=20kHz, Rs1

=2.5kHz, t01

=0, φ01

=0Singal 2: QPSK F

c2=20.1kHz, R

s2=2.4kHz, t

02=0.1,φ

02=2π*0.1485


TH=0.2

(a) 5 dB received power difference

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

alpha(Hz)

f(H

z)


Signal 1: BPSK Fc1

=20kHz, Rs1

=2.5kHz, t01

=0, φ01

=0Singal 2: QPSK F

c2=20.1kHz, R

s2=1.2kHz, t

02=0.1,φ

02=2π*0.7061


TH=0.2

(b) 10 dB received power difference

Figure 4.6: SOF of two different signals with different received power difference

Hence, let’s investigate the detection accuracy as a function of the relative power be-

tween the mixed signal components. The circle-shaped curvein Figure4.7 demonstrates

the detection accuracy versus the received signal power differences. Both signals are trans-

mitted through independent fading channels with 10 dB received SNR. If the algorithm

correctly detect the two signals and accurately estimate their symbol rates, it is counted as

a correct detection, otherwise it is deemed incorrect.

As can be seen from the figure, when the received power difference is less than 6 dB,

our method has excellent accuracy higher than 90%. As the difference increases to 10 dB,

the detection accuracy gradually falls to around 10%.

The star-shaped curve Figure4.7shows the detection/estimation result when SNR is 0

dB. Compared with the curve of 10 dB, we can find the SNR of channel has little effect on

the performance when received power differences is less then 4 dB. This is the direct result

of the noise resistance property of cyclostationary analysis, with noise having no spectrum

correlation. In order to increase the detection accuracy, we need to increase the length of

36

0 2 4 6 8 100

10

20

30

40

50

60

70

80

90

100


Det

ectio

n ac

cura

cy (

%)

Received−power differences vs. Detection accuracyfor mixed signal detection in Flat fading channel

2 components SNR = 10 dB2 components SNR = 0 dB3 components SNR = 10 dB

Figure 4.7: Received-power differences vs. Detection accuracy in Flat fading channel

the mixed signal, which will increase the complexity of calculation.

4.5.3 Mixed Signal with More Signal Components

The square-shaped curve in Figure4.7 shows detection performance for the case of three

modulated signal components in the mixed signal (1 BPSK and 2QPSK) and every two

adjacent components have 100% spectrum overlap, and the received power difference be-

tween the highest and the lowest components is 5 dB. It can be found that components’

number has little effect on the detection accuracy.

4.5.4 Threshold for Detection

In order to get a more reliable threshold value, we tried different values for decision. Figure

4.8shows the detection accuracy versus the received signal power differences for different

threshold value. As we can see, when the power difference is less than 6 dB, we can use

37

a higher threshold value, but when power difference is larger, we need to adjust to a lower

threshold. According to this, we are going to choose threshold value dynamically in future

research.

Figure 4.8: Received-power differences vs. Detection accuracy for different thresholdvalue

38

Conclusion

In this thesis, we employ second-order cyclostationary features to detect the number of sig-

nals in mixed-signals and demonstrate the effectiveness and efficiency of the second-order

cyclostationary feature based detection and RF parameter estimation algorithms. We also

develop a SOF based algorithm to detect mixed signals with significant spectrum overlap

and estimate their carrier frequencies. Because of the invariance property of SOF, we can

choose a universal threshold for detection. Since additivenoise has no spectrum correla-

tion, the proposed SOF based algorithm is resistant to noise. Experiments under different

channel conditions and signal mixture scenarios confirm theeffectiveness and robustness

of the proposed scheme.

For higher order modulation type, firstly, we attempted fourth-order cumulant based

mixed signal detection and then employ second-order cyclostationary features to perform

signal detection and symbol rate estimation in mixed signalin spectrally congested and

spectrally contested environment. By exploiting the SOF, we can detect the number of

signal components and estimate their symbol rate accurately in different channel conditions

and signal to noise levels. We also demonstrate the effectiveness and robustness of the

proposed method through simulation.

39

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