Mixed Signal Detection and Parameter Estimation Based on Second-Order Cyclostationary Features

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Dong Li Wright State University

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A thesis submitted in partial fulﬁllment of the requirements for the degree of Master of Science in Engineering

Dong Li B.S., Tianjin University, 2009

2015 Wright State University Wright State University GRADUATE SCHOOL

December 1, 2015

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPER- VISION BY Dong Li ENTITLED Mixed Signal Detection and Parameter Estimation based on Second-Order Cyclostationary Features BE ACCEPTED IN PARTIAL FUL- FILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in Engineering.

Zhiqiang Wu, Ph.D. Thesis Director

Brian Rigling, Ph.D. Department Chair

Committee on Final Examination

Zhiqiang Wu, Ph.D.

Kefu Xue, Ph.D.

Yan Zhuang, Ph.D.

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

Li, Dong. M.S.Egr., Department of Electrical Engineering, College of Engineering & Computer Science, Wright State University, 2015. Mixed Signal Detection and Parameter Estimation based on 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 sensing in 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, multiple signals are often mixed together at the signal detector with signiﬁcant overlap in spectrum. Conventional frequency analysis 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 mixed signals. 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 and signal mixture scenarios. Sim- ulation results conﬁrm the effectiveness of the proposed schemes.

iii List of Symbols Chapter 1

CR Cognitive Radio DSA Dynamic Spectrum Access RF Radio Frequency SCF Spectral Correlation Function SOF Spectrum Coherence Function Chapter 2

MISO Multiple Input Single Output E· Expected Value α Rx (τ) Cyclic Autocorrelation Function α Sx (f) Spectral Correlation Function α Cx (f) Spectrum Coherence Function C(kj) k-th order cumulant Chapter 3

TH Threshold Value N Number of components in mixed-signal fc Carrier Frequency Rs Symbol Rate B Null-to-Null bandwidth

iv Contents

1 Chapter 1: Introduction 1 1.1 Signal Detection, RF Parameter Estimation and Mixed Signal Detection/RF Parameter Estimation 1.2 Motivation ...... 2 1.3 Thesis Outline ...... 3

2 Chapter 2: Mixed Signal Detection and Second-Order Cyclostationary Theory 4 2.1 Signal Detection and RF Parameter Estimation ...... 4 2.2 Mixed Signal Detection ...... 5 2.3 Second-order Cyclostationary Features ...... 6 2.3.1 Cyclostationary Autocorrelation Function ...... 6 2.3.2 Spectral Correlation Function (SCF) ...... 7 2.3.3 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 9 3.1 Mixed-signal overlapping model ...... 9 3.2 Mixed-signal Detection based on SCF ...... 9 3.3 Performance of SCF based Mixed Signal Detection ...... 13 3.3.1 Estimate the Kratio Value ...... 13 3.3.2 Mixed Signal with Two Components ...... 14 3.3.3 Mixed Signal with Two Components in Different Channel Conditions 14 3.3.4 Mixed Signal with Two Components with Different Spectrum Overlap 15 3.3.5 Mixed Signal with More Components ...... 15 3.3.6 Mixed Signal with More Components in Different Noise Conditions 17 3.4 Mixed Signal Detection based on SOF ...... 18 3.5 Performance of SOF based mixed signal detection ...... 20 3.5.1 Comparison Between SOF and SCF based Algorithms ...... 20 3.5.2 Effects of Different Channel Conditions and Noises ...... 21 3.5.3 Resistance to Different Spectrum Overlap ...... 23 3.5.4 Mixed Signal with More Signal Components ...... 23

v 4 Chapter 4: Mixed Signal Detection for Higher Order Modulations 25 4.1 SCF and SOF for Mixed Signals with Higher-order Modulations ...... 25 4.2 Fourth-order Cumulant based method for detection ...... 26 4.3 SCF and SOF for M-QAM signal ...... 27 4.4 SOF based Mixed Signal Detection and Symbol Rate Estimation ...... 32 4.5 Performance of SOF based Mixed Signal Detection and Symbol Rate Estimation 34 4.5.1 Detection of Different Components with Different Modulation Types 34 4.5.2 Effects of Different Channel Conditions and Noises ...... 35 4.5.3 Mixed Signal with More Signal Components ...... 37 4.5.4 Threshold for Detection ...... 37

5 Conclusion 39

Bibliography 40

vi List of Figures

2.1 Signal Detection Model ...... 4 2.2 Spectrum of mixed signal with two different components ...... 5

3.1 Spectra of two signals with 100% spectrum overlap ...... 10 3.2 SCF of two different BPSK signals with 90% spectrum overlap ...... 11 3.3 The probability density function of Kratio ...... 13 3.4 SCF of two different BPSK Signals with 0% and 95% spectrum overlap .. 14 3.5 SCF of two different BPSK signals with 95% spectrum overlap in flat fading channel 15 3.6 SCF of two different BPSK signals with 99% spectrum overlap in flat fading channel 16 3.7 SCF of mixed BPSK signal with more than two components in 95% spectrum overlap 16 3.8 SCF of three different BPSK signals with 95% spectrum overlap in AWGN channel 17 3.9 SCF of three different BPSK signals with 95% spectrum overlap in Flat Fading channel 17 3.10 SOF of two different BPSK signals with 100% spectrum overlap ...... 20 3.11 SCF and SOF of two different BPSK signals with 7 dB received power difference 21 3.12 SOF of two different BPSK signals with different received power differences 22 3.13 Received-power differences vs. Detection accuracy for two components .. 23 3.14 SOF of two different BPSK signals with 7 dB received power difference .. 24 3.15 SOF and Detection accuracy for three components in flat fading channel .. 24

4.1 SCF of BPSK and QPSK signal ...... 26 4.2 C42 values for signal components with different Received Power Differences 27 4.3 SOF of different modulated signal when f = fc and α ∈ (0, 1.2Rs) .... 30 4.4 SOF of two different signal components with 100% spectrum overlap ... 34 4.5 SOF of two signals with 0 dB received power difference ...... 35 4.6 SOF of two different signals with different received power difference .... 36 4.7 Received-power differences vs. Detection accuracy in Flat fading channel . 37 4.8 Received-power differences vs. Detection accuracy for different threshold value 38

vii Chapter 1: Introduction

1.1 Signal Detection, RF ParameterEstimationand 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, interference 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 and secondary 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 identification 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 signal without 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 signals at slightly different carrier 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 identifying 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 coherence function (SOF) features [10–16]. We demonstrate the effectiveness and efﬁciency of the second-order cyclostationary features under different conditions. Furthermore, we develop a sophisticated robust algorithm to detect mixed signals and estimate their RF parameters when higher order modulations such 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, RF parameter 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 signal 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 cyclostationary feature based mixed signal detection algorithm and carrier 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 cumulant and propose a new method for higher-order modulation type. 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 communication, radar, navigation, and electronic warfare. Figure 2.1 shows 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 primary users (PUs) and spectrum holes. It has also been realized that to ensure more sophisticated cognitive radio systems, important RF parameters such as carrier frequency, symbol rate, and modulation type need to be detected as well.

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4 2.2 Mixed Signal Detection

However, in most existing work, the target signal is often assumed to be a single communication signal without overlap with other signals in the frequency domain. Figure 2.2(a) shows the spectrum of two narrow-band BPSK signal components without 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 spectrum 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 Figure 2.2(b) that we can no longer tell that there are two signal components by simple frequency analysis, and we cannot use a narrow-band filter to distinguish one signal from the other. This is what we are trying to achieve in this thesis: use cyclostationary analysis to determine how many signal components exist and estimate their parameters.

Spectrum Analysis of mixed signals Component 1: QPSK signal (Fc=15000, SymbolRate=2500) (Hz) Spectrum Analysis of mixed signals Component 2: BPSK signal (Fc=25000, SymbolRate=2400) (Hz) Component 1: QPSK signal (Fc = 20000, SymbolRate = 2500) (Hz) Spectrum Overlap: 0 % Component 2: BPSK signal (Fc = 20100, SymbolRate = 2400) (Hz) 20 Spectrum Overlap: 100 % 30 18

16 25

14 20 12

10 15 Amplitude

8 Amplitude 10 6

4 5 2

0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 4 4 Frequency (Hz) x 10 Frequency (Hz) x 10 (a) 0% spectrum overlap (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 processes. 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 spectrum. By employing cyclostationary analysis, e.g., using the correlation characteristics between 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

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

τ τ τ τ R (t + , t − )= E[x(t + )x∗(t − )], (2.1) x 2 2 2 2

τ τ Here, Rx(t + 2 , t − 2 ) is a periodic function with period T . Decompose this signal into Fourier Series, we have

τ τ R (t + , t − )= Rα(τ)ej2παt (2.2) x 2 2 x Xα where the Fourier coefﬁcients are

1 T/2 τ τ Rα(τ) = lim R (t + , t − )e−j2παtdt (2.3) x →∞ x T T Z−T/2 2 2

α where α = m/T is the cyclic frequency, Rx (τ) is the cyclic autocorrelation function.

6 2.3.2 Spectral Correlation Function (SCF)

α The Fourier transformation of cyclic autocorrelation function Rx (τ) is named Spectral Cor- relation Function(SCF), which is presented as

∞ α α −j2πfτ Sx (f)= Rx (τ)e dτ. (2.4) Z−∞

where α 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 detection 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 solvethis problem. Similar to SCF, SOF is a two variable (frequency nf and cyclic frequency α) function. Assume x(t) is a cyclostationary signal, its SOF defined in [10–15] is:

α α Sx (f) Cx (f)= ∗ , (2.5) 1 1 1/2 S0 f + α S0 f − α x 2 x 2

It is shown that the SOF is upper-bounded by unity (shown as Equation 2.6) and can help to remove channel effect [11]:

α |CX (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 and 16-QAM signals) have different 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 process y(n) are deﬁned as [19, 20]:

1 N Cˆ = |y(n)|2 (2.7) 20 N Xn=1 1 N Cˆ = y2(n) (2.8) 21 N Xn=1 1 N Cˆ = y4(n) − 3Cˆ2 (2.9) 40 N 20 Xn=1 1 N Cˆ = y3(n)y∗(n) − 3Cˆ Cˆ (2.10) 41 N 20 21 Xn=1 1 N Cˆ = |y(n)|4 −|Cˆ |2 − 2Cˆ2 (2.11) 42 N 20 21 Xn=1 where E[y(n)]=0. In [1], we used the normalized fourth-order cumulant deﬁned in Equation 2.12 to distinguish different modulation types:

Cˆ4k Cˆ4k = , k =0, 1, 2 (2.12) ˆ2 C21

The theoretical normalized Cˆ42 for QPSK and 16QAM signals are [19]:

Cˆ42(QPSK) = −1.0000

Cˆ42(16QAM) = −0.6047

8 Chapter 3: Mixed-signal Detection and

Carrier Frequency Estimation

3.1 Mixed-signal overlapping model

Figure 3.1 shows an example of mixed signal detection with signiﬁcant overlap. It is clear from Figure 3.1 that the second signal’s spectrum is entirely contained within the ﬁrst signal’s spectrum with a 100% overlap. This is the worst case scenario for spectrum overlapping. 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 signals with slightly different carrier frequencies:

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

where a1(t) and a2(t) are the envelope of the two individual signals, f1 and f2 are the

9 !" #

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

carrier frequencies which are very close, and φ1 and φ2 are the phases of the two signals. ˆα Equation 3.2 shows that when α = ±2fc1 and α = ±2fc2 , Sa (f) has maximum value [9].

1 ˆα ˆα ˆα ˆα Sa1 (f + f1)+ Sa1 (f − f1)+ Sa2 (f + f2)+ Sa2 (f − f2) , α =0  4 h i  1 − 1 1  Sˆα 2f (f)e+j2φ , α = +2f  4 a1 1  1  ˆα+2f1 −j2φ1  Sa1 (f)e , α = −2f1 ˆα  4 Sx (f)=  (3.2)  1 − 2 2  Sˆα 2f (f)e+j2φ , α = +2f 4 a2 2 1  ˆα+2f2 +j2φ2  Sa2 (f)e , α = −2f2  4   0, otherwise    According to this feature, we can now easily identify how many signals are co-existing in the mixed signal. The following low complexity algorithm summarizes the procedure of our mixed signal detection:

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

carrier frequency fc and bandwidth.

2. Next, we can set an appropriate scope of α values in order to reduce the amount of computation. However, in order to obtain the accurate value of 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 then be estimated via second- order cyclostationary features.

Spectral correlation magnitude features for mixed signals

6 x 10 2 BPSK signals with Spectrum Overlap: 90 % 2.5 P1 P2 alpha=−8220 Hz 2 alpha=−16220 Hz

H H 1.5 win 2 win 2 SCF win 1 1

0.5

0 −2Fc2 −2Fc1 α

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

Figure 3.2 shows 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 value P1 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. Value H, 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 value P2 in the rest area ’win2’.

5. Compare P1 and P2, and deﬁne a ratio value ’Kratio’ using Equation 3.3

P 2 Kratio = (3.3) P 1

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 value TH

TH = P 1 × Kratio (3.4)

8. Make a decision of the signal number N of the mixed signal using a threshold value TH

2, if two values of SCF >TH N =  (3.5)  1, if one values of SCF >TH  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 Equation 3.4

12 3. Make decision using threshold value TH

Number of signals = N

where N 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

The probability density function of kratio for two signals and one signal 12 2 signals (20 dB 1:0.5) 1 signal (20 dB) K 10

6 PDF

0 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 kratio value

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 Figure 3.3. 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 channel is 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 Figure 3.3 that the Kratio value should be set to K = 0.27. After we ﬁnd the maximum peak P1, we then use K to calculate the threshold 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 Figure 3.4(a), the signal is composed of two different BPSK signals with 0% spectrum overlap. In comparison, Figure 3.4(b) shows the SCF of a mixed signal, which is composed

of two BPSK signals with different carrier frequencies with 95% spectrum overlap. It is

evident that we can ﬁnd that there are two peaks at α = −2fc1 and α = −2fc2 .

Spectral correlation magnitude features for mixed signals Spectral correlation magnitude features for mixed signals

6 6 x 10 2 BPSK signals with Spectrum Overlap: 0 % x 10 2 BPSK signals with Spectrum Overlap: 95 % 6 6

alpha=−40000 Hz

5 alpha=−40000 Hz 5 alpha=−42000 Hz alpha=−80000 Hz

4 4

3 3 SCF SCF

2 2

1 1

0 0 −2Fc2 −2Fc1 −2Fc2 −2Fc1 α (Hz) α (Hz)

(a) 0% spectrum overlap (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

Figure 3.5 shows the SCF when the mixed signals are under fading. The fading factor for

component 1 is α1 =1, and for component 2 is α2 =0.5. The SNR for each component in the mixed signals are 20 dB (in Figure 3.5(a)) and 10 dB (in Figure 3.5(b)).

From these ﬁgures, it is evident that the proposed SCF based mixed signal detection algorithm is robust to channel conditions and noises.

14 Spectral correlation magnitude features for mixed signals Spectral correlation magnitude features for mixed signals (In Flat Fading Channe: SNR = 20 dB) (In Flat Fading Channe: SNR = 10 dB) 6 6 x 10 2 BPSK signals with Spectrum Overlap: 95 % x 10 2 BPSK signals with Spectrum Overlap: 95 % 6 5 alpha=−40000 Hz 4.5 alpha=−40000 Hz 5 4

3.5 4 3

3 2.5 SCF SCF 2 2 1.5 alpha=−42000 Hz alpha=−42000 Hz 1 1 0.5

0 0 −2Fc2 −2Fc1 −2Fc2 −2Fc1 α (Hz) α (Hz)

(a) SNR= 20dB (b) SNR= 10dB

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

3.3.4 Mixed Signal with Two Components with Different Spectrum

Overlap

Figure 3.6 shows the SCF of the scenario when two components of the mixed signal have larger spectrum overlap (99% overlap). The signal is transmitted in a ﬂat fading channel.

The fading factor for component 1 is α1 = 1, and for component 2 is α2 = 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. Figure 3.7(a) shows the SCF features for mixed signal with three spectrum components. We can clearly observe the three peaks in the SCF. Figure 3.7(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 Spectral correlation magnitude features for mixed signals (In Flat Fading Channe: SNR = 20 dB) 6 x 10 2 BPSK signals with Spectrum Overlap: 99 % 5 alpha=−40000 Hz 4.5

3.5

2.5 SCF 2

1.5 alpha=−40400 Hz 1

0.5

0 −2Fc2 −2Fc1 α (Hz)

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

Spectral correlation magnitude features for mixed signals Spectral correlation magnitude features for mixed signals

6 6 x 10 3 BPSK signals with Spectrum Overlap: 95 % x 10 5 BPSK signals with Spectrum Overlap: 95 % 6 7

alpha=−44000 Hz alpha=−44000 Hz 6 alpha=−48000 Hz 5 alpha=−40000 Hz alpha=−42000 Hz 5 alpha=−46000 Hz 4 alpha=−40000 Hz

alpha=−42000 Hz 4 3 SCF SCF 3

2 2

1 1

0 0 −2Fc3 −2Fc2 −2Fc1 −2Fc5 −2Fc4 −2Fc3 −2Fc2 −2Fc1 α (Hz) α (Hz)

(a) Mixed signal with 3 components (b) Mixed signal with 5 components

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

16 Spectral correlation magnitude features for mixed signals Spectral correlation magnitude features for mixed signals (In AWGN Channe: SNR = 20 dB) (In AWGN Channe: SNR = 10 dB) 6 6 x 10 3 BPSK signals with Spectrum Overlap: 95 % x 10 3 BPSK signals with Spectrum Overlap: 95 % 6 5 alpha=−42000 Hz 4.5 alpha=−40000 Hz 5 alpha=−44000 Hz 4 alpha=−42000 Hz

alpha=−44000 Hz 3.5 4 alpha=−40000 Hz 3

3 2.5 SCF SCF 2 2 1.5

1 1 0.5

0 0 −2Fc3 −2Fc2 −2Fc1 −2Fc3 −2Fc2 −2Fc1 α (Hz) α (Hz)

(a) SNR= 20dB (b) SNR= 10dB

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

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 conditions. Figure 3.8 shows the case where the signals are transmitted in AWGN channels. Each component in the mixed signals has the same SNR, which is 20 dB (in Figure 3.8(a)) and 10 dB (in Figure 3.8(b)). We can see that in AWGN channel, the noises have little effects on the SCF features.

Spectral correlation magnitude features for mixed signals Spectral correlation magnitude features for mixed signals (In Flat Fading Channe: SNR = 20 dB) (In Flat Fading Channe: SNR = 10 dB) 6 6 x 10 3 BPSK signals with Spectrum Overlap: 95 % x 10 3 BPSK signals with Spectrum Overlap: 95 % 4.5 6 alpha=−40000 Hz 4 5 alpha=−40000 Hz 3.5 alpha=−42000 Hz 3 4 alpha=−42000 Hz 2.5 3 SCF 2 SCF alpha=−44000 Hz 1.5 2 alpha=−44000 Hz 1 1 0.5

0 0 −2Fc3 −2Fc2 −2Fc1 −2Fc3 −2Fc2 −2Fc1 α (Hz) α (Hz)

(a) SNR= 20dB (b) SNR= 10dB

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

17 Figure 3.9 shows the case where the signals are transmitted in ﬂat fading channels.

The fading factors for different signal components are different, which are α1 : α2 : α3 = 1:0.8:0.6. Each component in the mixed signal has the same SNR, which is 20 dB (in Figure 3.9(a)) and 10 dB (in Figure 3.9(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 provide a ﬁxed threshold value for detection and a heuristic algorithm needs to be performed to ﬁnd appropriate detection threshold.

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

ized version of SCF to solve this problem. Assume x(t) is a cyclostationary signal, its SOF is deﬁned in Equation 2.5. We have known that the SOF is upper-bounded by unity and can help to remove channel effect [11].

Here, we just focus on f = 0 projection of the SOF because for typical communication signals, the SOF exhibits special features in the cyclic frequency domain when f =0. Assume the received mixed signal is composed by two independent BPSK signals with carrier frequencies fc1 and fc2 , 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.

α α After normalization of Sa (f) using Equation 2.5, we obtain the SOF Ca (f). Next, we

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

1. Estimate the mixed signal’s carrier frequency fˆc and bandwidth Bˆ through spectrum analysis.

2. Calculate the SOF around 2fˆc for a cyclic frequency span of Bˆ with ﬁne resolution.

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

ˆ The ﬁrst step is a simple spectrum analysis and the carrier frequency estimation fc is not very accurate. This estimation is only used for the second step to perform high resolution 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. Figure 3.10 shows 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 to Rs1 = 2500 Hz and Rs2 = 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 dB different from each other. It is evident from Figure 3.10 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 of 0.2 and count the number of peaks above this threshold, then estimate the carrier frequency from the cyclic frequencies of such peaks.

19 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*0 Signal : BPSK Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.24565 0.35 alpha=40000 Hz P1

0.3 P2 alpha=40200 Hz 0.25

0.2 TH = 0.2 SOF 0.15

0.1

0.05

2Fc1 2Fc2 α (Hz)

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. Figure 3.11 show 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 first one. The SNR of mixed- signal is set to be 10 dB (here we define SNR as the ratio between total 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 Spectral correlation magnitude features for mixed signals Spectral coherence features for mixed signals (In Flat Fading Channe: SNR = 10 dB) (In Flat Fading Channe: SNR = 10 dB) 2 signals with Spectrum Overlap: 81.25 % 2 signals with Spectrum Overlap: 81.25 % Signal : BPSK Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0 Signal : BPSK Power = 1 Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0 Signal6 : BPSK Fc2=21000 Fb2=2400 tau2=0.1 phy2=2*pi*0.55461 x 10 Signal : BPSK Power = 0.19953 Fc2=21000 Fb2=2400 tau2=0.1 phy2=2*pi*0.55461 alpha=40000 Hz 0.9 5 alpha=40000 Hz

4.5 0.8 4 0.7

3.5 0.6 3 0.5 SCF

2.5 SOF 0.4 2 0.3 alpha=42000 Hz 1.5 0.2 1 alpha=42000 Hz 0.5 0.1

2Fc1 2Fc2 2Fc1 2Fc2 α (Hz) α (Hz)

(a) SCF (b) SOF

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

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 transmitted 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 different 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 parameters for 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 Figure 3.12(b), the difference of received power between the two components is 5 dB; and in Figure 3.12(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 Spectral coherence features for mixed signals Spectral coherence features for mixed signals (In Flat Fading Channe: SNR = 10 dB) (In Flat Fading Channe: SNR = 10 dB) 2 signals with Spectrum Overlap: 100 % 2 signals with Spectrum Overlap: 100 % Signal : BPSK Power = 1 Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0 Signal : BPSK Power = 1 Fc1=20000 Fb1=2500 tau1=0 phy1=2*pi*0 Signal : BPSK Power = 1 Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.77622 Signal : BPSK Power = 0.31623 Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.68806 0.55 alpha=40000 Hz alpha=40200 Hz 0.5 alpha=40000 Hz 0.7

0.45 0.6 0.4 0.5 0.35

0.3 0.4

SOF 0.25 SOF 0.3 0.2 alpha=40200 Hz

0.15 0.2 0.1 0.1 0.05

2Fc1 2Fc2 2Fc1 2Fc2 α (Hz) α (Hz)

(a) 0 dB received power difference (b) 5 dB received power difference

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*0 Signal : BPSK Power = 0.1 Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.30703 0.9 alpha=40000 Hz 0.8

0.7

0.6

0.5

SOF 0.4

0.3

0.2 alpha=40200 Hz 0.1

2Fc1 2Fc2 α (Hz)

Figure 3.12: SOF of two different BPSK signals with different received power differences tween the mixed signal components. Figure 3.13(a) demonstrates the detection accuracy versus the received signal power differences. Both signals are transmitted through independent 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 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 which is higher than 90%. As the difference increases to 10 dB, the detection accuracy gradually falls to around 55%. Figure 3.13(b) shows the detection/estimation result when SNR is 0 dB. Compared with Figure 3.13(a), we ﬁnd that the SNR of channel has little effect on the performance.

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

22 Received−power differences vs. Detection accuracy Received−power differences vs. Detection accuracy for two components in Flat fading channel (SNR = 10dB) for two components in Flat fading channel (SNR = 0dB) 100 100

90 90

80 80

70 70

60 60

50 50

40 40

Detection accuracy (%) 30 Detection accuracy (%) 30

20 20

10 10

0 0 0 2 4 6 8 10 0 2 4 6 8 10 Received−power differences between two components (dB) Received−power differences between two components (dB)

(a) SNR= 10dB (b) SNR=0dB

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 spectrum overlap 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 Figure 3.11(b), it is observed that the performances are almost the same. This indicates that the proposed mixed signal detection and carrier frequency estimation algorithm is robust to signiﬁcant spectrum overlap.

3.5.4 Mixed Signal with More Signal Components

Figure 3.15(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 below 90%.

23 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*0 Signal : BPSK Power = 0.19953 Fc2=20100 Fb2=2400 tau2=0.1 phy2=2*pi*0.63807 0.8 alpha=40000 Hz

0.7

0.6

0.5

0.4 SOF

0.3 alpha=40200 Hz 0.2

0.1

2Fc1 2Fc2 α (Hz)

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

Received−power differences vs. Detection accuracy SOF features for mixed signals for 3 components in Flat fading channel (SNR = 5dB) (In Flat Fading Channe: SNR = 10 dB) 100 3 signals with Spectrum Overlap: 100 % 0.55 alpha=40000 Hz 90 0.5 80 0.45 70 0.4 60 0.35

0.3 alpha=40200 Hz 50 SOF 0.25 40 alpha=40400 Hz 0.2 Detection accuracy (%) 30 0.15 20 0.1 10 0.05 0 2Fc1 2Fc2 2Fc3 0 1 2 3 4 5 6 α (Hz) largest Received−power differences among 3 components (dB)

(a) SOF of three different BPSK signals with 100% (b) largest Received-power differences vs. Detection spectrum overlap accuracy 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 signals with only BPSK components. Now let’s expand our algorithm to include higher modulation 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 at α = 2fc in the cyclostationary domain(shown in Figure 4.1) [14]. As a direct result, we can no longer use previous method to detect the 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. Figure 4.2 is the

simulation result which shows the C42 value of different combinations. The x-axis shows the changing of received power differences between two components. As can be seen from Figure 4.2, we can not decide a ﬁxed threshold value to distinguish different curves. The conclusion can be drawn is that the fourth-order cumulant method does not work for mixed signal detection.

26 C42 values for different combination of signal components in different Received Power Differences 1.5

BPSK & BPSK 0.5 BPSK & QPSK BPSK & 16QAM QPSK & QPSK QPSK & 16QAM C42 values 0 16QAM & 16QAM

−0.5

−1 0 2 4 6 8 10 Received Power Differences (in dB)

Figure 4.2: C42 values for signal components with different Received Power Differences

4.3 SCF and SOF for M-QAM signal

According to Figure 4.1, we noticed that both BPSK and QPSK signal have a highest

side-lobe at f = 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-series x(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 of x(t) is:

1 1 Rα(τ)= [Rα(τ)+ Rα(τ)] cos(2πf τ)+ [Rα (τ) − Rα (τ)] sin(2πf τ) x 2 c s c 2 sc cs c 1 (4.2) + [Rα+2nfc (τ) − Rα+2nfc (τ)] + ni[Rα+2nfc (τ)+ Rα+2nfc (τ)] , 4 c s sc cs n=X−1,1 and the SCF of x(t) is:

1 Sα(f)= {[Sα(f + nf )+ Sα(f + nf )] + ni[Sα (f + nf ) − Sα (f + nf )]} x 4 c c s c sc c cs c n=X−1,1 1 + [Sα+2nfc (f) − Sα+2nfc (f)] + ni[Sα+2nfc (f)+ Sα+2nfc (f)] . 4 c s sc cs n=X−1,1 (4.3)

Here, c(t) and s(t) are cyclostationary. For the special case in which the in-phase and quadrature components c(t) and s(t) are joint purely stationary, x(t) is purely cyclostationary with period T0 =1/2fc, and we have:

1 Sx(f)= [Sc(f + fc)+ Sc(f − fc)+ Ss(f + fc)+ Ss(f − fc)] 4 (4.4) 1 − [S (f + f ) − S (f − f ) ] 2 cs c i sc c i and 1 1 Sα(f)= [S (f) − S (f)] ± iS (f) , α = ±2f x 4 c s 2 cs r c

α with Sx =0 for α =06 and α =26 fc. In digital QAM modulation, c(t) and s(t) are PAM signals [11], which are

∞

c(t)= cnq(t − nT0 − t0), X−∞ ∞

s(t)= snq(t − nT0 − t0), X−∞

28 For c(t), we have:

1 α α α ∗ Q(f + )Scn (f + )Q(f − ) , α = k/T0 α  T0 2 2 2 Sc (f)= (4.5)  0, α =6 k/T0  in which sin(πfT ) Q(f)= 0 , πf

For bits sequence cn, we have

˜ Rcn (0), α = k/T0 S˜α (f)=  , |k| =0, 1, 2, ... cn   0, α =6 k/T0  ˜ here, we assume Rcn (0) = 1, and k is an integer.

According to the spectral correlation function, if we focus on f = fc projection of the SCF, we can derive that:

T k Sˆα(f )= 0 sinc2( ), α = k/T , |k| =0, 1, 2, ... (4.6) x c M 2 0 where sin(πx) sinc(x)= . πx

ˆα Hence, Sx (fc) shows a ﬁxed feature at f = fc and α = k/T0 (shown in Figure 4.1), which can be used for our purpose of mixed signal component detection.

According to Equation 2.5, we have:

α Cx (fc)=1, when α = k/T0. (4.7)

The SOF for BPSK, QPSK, and 16-QAM are shown in Figure 4.3. Here, we set the

29 carrier frequency fc = 20000 Hz, the null-to-null bandwidth of the signal B = 5000 Hz, which means the symbol rate Rs = 2500 Hz. The time delay t0 and phase offset φ0 are set to be 0. Here, we just focus on f = fc and α ∈ (0, 1.2Rs) projection.

SOF for signal SOF for signal Signal : BPSK Fc=20kHz, SymbolRate=2.5kHz Signal : QPSK Fc=20kHz, SymbolRate=2.5kHz 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 f(Hz) f(Hz) 0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 alpha(Hz) alpha(Hz)

(a) BPSK (b) QPSK

SOF for signal Signal : 16QAM Fc=20kHz, SymbolRate=2.5kHz 1

0.9

0.8

0.7

0.6

f(Hz) 0.5

0.4

0.3

0.2

0.1 0 500 1000 1500 2000 2500 3000 alpha(Hz)

Figure 4.3: SOF of different modulated signal when f = fc and α ∈ (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), X−∞

30 we have

T k Sˆα(f ) = 0 sinc2( ), α = k/T ,, |k| =0, 1, 2, ... (4.8) x c (BPSK) 2 2 0

The SOF of a BPSK signal are shown in Figure 4.3(a).

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

∞

c(t)= cnq(t − nT0 − t0), X−∞ ∞

s(t)= snq(t − nT0 − t0), X−∞

Similarly, if we focus on f = fc projection of the SCF and SOF, we can derive out that for QPSK we have

T k Sˆα(f ) = 0 sinc2( ), α = k/T . (4.9) x c (QPSK) 4 2 0

The SOF of QPSK signal is shown in Figure 4.3(b). Similarly, we set the carrier

frequency fc = 20000 Hz, the symbol rate of QPSK Rs = 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 signal x(t) is composed by two independent signals x1(t)

and x2(t) with carrier frequencies fc1 and fc2 , 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:

ˆα ˆα ˆα Sx (f)= Sx1 (f)+ Sx2 (f). (4.10)

If we focus on f = fc projection, we will ﬁnd that when α = kRs(i), where Rs(i)= ˆα 1/T0(i) is the symbol rate of i-th component, Sx (fc) exhibits an impulse which can be exploited to detect the existence of multiple components of each and every component.

α α After normalization of Sx (f) using Equation 2.5, we can obtain the SOF Cx (f). At the same time, because SCF (or SOF) is a two variable (f and α) function, and the carrier frequencies of the two components are different, we just focus on the band

f ∈ (fˆc − B/ˆ 2, fˆc + B/ˆ 2) to detect the multiple components. Next, we use the following three-step algorithm to detect signals and estimate carrier frequencies.

32 ˆ 1. Step 1: Estimate the mixed signal’s carrier frequency fc and null-to-null bandwidth Bˆ through spectrum analysis.

2. Step 2: Calculate the SOF around B/ˆ 2 for a cyclic frequency span of Bˆ with ﬁne resolution.

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

ˆ The ﬁrst step is a simple spectrum analysis and the carrier frequency estimation fc 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. Figure 4.4(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 and fc2 = 20100 Hz. The null-to-null bandwidth are set to B1 = 5000

Hz and B2 = 4800 Hz, which means the symbol rates are Rs1 = 2500 Hz and Rs2 = 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 dB different from

each other.

Figure 4.4(b) is the view of Figure 4.4(a) from α axis. It is evident from Figure 4.4(b) that the two signal components exhibit peaks at k 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 of 0.2 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 Figure 4.4(b), there are 2 peaks at cyclic frequency α = 2400, 2500

33 Spectral coherence features for mixed signals (In Flat Fading Channel: SNR = 10 dB, power difference = 3 dB) Signal 1: BPSK F (1)=20kHz, R (1)=2.5kHz, t (1)=0, φ (1)=0 c s 0 0 Singal 2: QPSK F (2)=20.1kHz, R (2)=2.4kHz, t (2)=0.1,φ (2)=2π*0.9056 c s 0 0 Spectrum Overlap: 100 %

0.7

0.6 X: 2500 Y: 0.6213

0.5

0.4 f(Hz)

X: 2400 0.3 Y: 0.3652

TH=0.2 0.2

0.1

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 alpha(Hz)

(a) 3-D view (b) α view

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

Hz, according to Equation 4.7, hence there are 2 components in this mixed signal, and their symbol rates are 2400 Hz and 2500 Hz.

4.5 Performanceof SOFbasedMixed SignalDetectionand

Symbol Rate Estimation

In this section, we evaluate the performance of the proposed mixed 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 with that proposed in our previous work is that, we extend our component’s number detection from only BPSK modulation to more modulation types (M-QAM). Figure 4.5(a) shows the SOF of two QPSK

34 signal components and Figure 4.5(b) shows that of two 16QAM components. As we can see from the ﬁgures, our method can detect different components with different modulation types.

Spectral coherence features for mixed signals Spectral coherence features for mixed signals (In AWGN Channel: SNR = 10 dB) (In Flat Fading Channel: SNR = 10 dB, power difference = 0 dB) Signal 1: QPSK F =20kHz, R =2.5kHz, t =0, φ =0 Signal 1: 16QAM F =20kHz, R =2.5kHz, t =0, φ =0 c1 s1 01 01 c1 s1 01 01 Singal 2: QPSK F =20.1kHz, R =2.4kHz, t =0.1,φ =1.4π Singal 2: 16QAM F =20.1kHz, R =1.2kHz, t =0.1,φ =2π*0.7061 c2 s2 02 02 c2 s2 02 02 Spectrum Overlap: 100 % Spectrum Overlap: 100 %

0.7 0.9

0.8 0.6 0.7

0.5 X: 2500 X: 2400 Y: 0.5233 0.6 Y: 0.502 0.4 0.5 f(Hz) f(Hz)

0.4 0.3

0.3 TH=0.2 0.2 TH=0.2 0.2

0.1 0.1

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 alpha(Hz) alpha(Hz)

(a) two QPSK signals (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 this method is the difference of

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

different channel conditions. Figure 4.6 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 these experiments are set to be the same except the relative power of the two components. In Figure 4.6(a), the difference of received power between the two components is 5 dB; and in Figure 4.6(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 Figure 4.6(b), the detection algorithm fails to detect the weak signal component.

Spectral coherence features for mixed signals Spectral coherence features for mixed signals (In Flat Fading Channel: SNR = 10 dB, power difference = 5 dB) (In Flat Fading Channel: SNR = 10 dB, power difference = 10 dB) Signal 1: BPSK F =20kHz, R =2.5kHz, t =0, φ =0 Signal 1: BPSK F =20kHz, R =2.5kHz, t =0, φ =0 c1 s1 01 01 c1 s1 01 01 Singal 2: QPSK F =20.1kHz, R =2.4kHz, t =0.1,φ =2π*0.1485 Singal 2: QPSK F =20.1kHz, R =1.2kHz, t =0.1,φ =2π*0.7061 c2 s2 02 02 c2 s2 02 02 Spectrum Overlap: 100 % Spectrum Overlap: 100 %

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5 f(Hz)

0.4 0.4

0.3 0.3 TH=0.2 TH=0.2 0.2 0.2

0.1 0.1

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 alpha(Hz) alpha(Hz)

(a) 5 dB received power difference (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 between the mixed signal components. The circle-shaped curve in Figure 4.7 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 Figure 4.7 shows 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 Received−power differences vs. Detection accuracy for mixed signal detection in Flat fading channel 100

Detection accuracy (%) 30

20 2 components SNR = 10 dB 10 2 components SNR = 0 dB 3 components SNR = 10 dB 0 0 2 4 6 8 10 Received−power differences between two components (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 Figure 4.7 shows detection performance for the case of three modulated signal components in the mixed signal (1 BPSK and 2 QPSK) and every two adjacent components have 100% spectrum overlap, and the received power difference between 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

4.8 shows 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 threshold value

38 Conclusion

In this thesis, we employ second-order cyclostationary features to detect the number of signals 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 additive noise has no spectrum correlation, the proposed SOF based algorithm is resistant to noise. Experiments under different channel conditions and signal mixture scenarios confirm the effectiveness 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 signal in 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.

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