Karlsruhe Institute of Technology Communications Engineering Lab Univ.-Prof. Dr.rer.nat. Friedrich K. Jondral

Transcription

1 Karlsruhe Institute of Technology Communications Engineering Lab Univ.-Prof. Dr.rer.nat. Friedrich K. Jondral Deployment of Energy Detector for Cognitive Relay with Multiple Antennas Bachelor Thesis by Lucas Rode s Guirao Supervisor : Advisor : Univ.-Prof. Dr.rer.nat. Friedrich K. Jondral Prof. Oriol Sallent M.Sc. Ankit Kaushik Begin End : : CEL

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3 Declaration I hereby declare that this thesis contains no material which has been accepted for award of any other degree or diploma at any university or equivalent institution and that, to best of my knowledge, the thesis contains no material previously published or written by an other person, except where due reference is provided in the text of the thesis. A complete list of references is included. Karlsruhe, the Lucas Rodes Guirao iii

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5 Abstract Key-words: Cognitive Radio, Cognitive Relay, Energy Detection, Square-law Combiner, GNU Radio Companion, USRP B210, Software Defined Radio. Deployment of Energy Detector for Cognitive Relay with Multiple Antennas is the title of Lucas Rodés bachelor thesis, submitted on 28th of July This project has been carried out at the Communications Engineering Lab (CEL) department of the Karlsruhe Institute of Technology (KIT) as an Erasmus+ program. Furthermore, it has been developed under the supervision of both Prof. Friedrich K. Jondral (Communications Engineering Lab, KIT, Karlsruhe) and Prof. Oriol Sallent (Mobile Communications Research Group, UPC Barcelona). The thesis advisor has been M.Sc. Ankit Kaushik (Communications Engineering Lab, KIT, Karlsruhe). Future technologies intend to redefine the spectrum access, making its usage more efficient and, by doing so, allowing new incoming wireless communications. This goal can be achieved by exploiting the degrees of freedom in time, frequency and spatial domain. For instance, to reuse the non-allocated spectrum in an opportunistic way (Interweave system), allowing primary and secondary usage of the same band spectrum. Motivated by these facts, Cognitive Relay (CR), a small cell deployment that provides coverage to secondary indoor devices, is proposed. A CR is meant to sense the licensed radio-electric spectrum and to determine if secondary users can access it. Following the progress in multiple antennas techniques, antenna diversity can be used to boost and improve the detector s performance in terms of receiver operating characteristics (probability of detection and probability of false alarm). This thesis studies the performance improvement, i.e. diversity gain, of an Square-Law Combiner (SLC) in Energy Detection. In addition, the theoretical developments are validated through hardware implementation. v

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7 Contents 1 Introduction 1 2 System Setup 7 1 AWGN Channel Fading Channel Single Antenna 15 1 AWGN Channel Hypothesis Hypothesis Performance Analysis Fading Channel Probability Density Function of Test Statistic Performance Analysis Square-Law Combiner 29 1 AWGN Channel Hypothesis Hypothesis Performance Analysis Fading Channel Probability Density Function of Test Statistic Performance Analysis Summary and Future Scope 41 vii

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9 Abbreviations Acronym AWGN CDF CLT CR DC DSA ED EGC GNU GUI IEEE IQ ISM LAN LOS MNO NLOS PDF PSD PSK PT ROC RV SINR SLC SLD SNR SP ST SU UHD USRP Description Additive White Gaussian Noise Cumulative Density Function Central Limit Theorem Cognitive Relay Direct Current Dynamic Spectrum Access Energy Detector Equal Gain Combining GNU s Not Linux Graphical User Interface Institute of Electrical and Electronics Engineers In-phase/Quadrature Industrial, Scientific and Medical Local Area Network Line-Of-Sight Mobile Network Operator Non-Line-Of-Sight Probability Density Function Power Spectral Density Phase-Shift Keying Primary Transmitter Receiver Operating Curve Random Variable Signal-to-Interference-plus-Noise Ratio Square-Law Combiner Square-Law Device Signal-to-Noise Ratio Signal Processing Secondary Transmitter Secondary User Universal Hardware Driver Universal Software Radio Peripheral ix

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11 Notation Notation Description B A certain spectrum band y[n] Processed received/sensed signal w[n] Processed received AWGN signal h Fading complex value α Path-Loss factor x[n] Processed PU received signal H 0 Hypothesis 0, absence of the PU H 1 Hypothesis 1, presence of the PU N Number of samples required to compute one Test Statistic value T (y) Test Statistic (single antenna case) λ Decision threshold of the binary hypothesis P fa Probability of False Alarm P d Probability of Detection P Transmitted PU power f c Carrier frequency of the PU signal f cent Center frequency of the PU signal x RF [n] Received/sensed, discretized and demodulated PU signal f s Sampling frequency F c Discrete carrier frequency B f Bandwidth of the bandpass filter used at the SU γ SNR at the SU in db M Decimation factor F ɛ Residual frequency of the PU sensed signal after processing it at the SU x(t) PU transmitted signal S Total samples of the sensed signal y[n] N coh Coherence time in number of samples T coh Coherence time in seconds [s] R y 2 Autocorrelation of the energy samples signal σw 2 Noise variance (single antenna case) CN (µ, σ 2 ) Complex Normal distribution with mean µ and variance σ 2 y R [n] = Re{y[n]} y I [n] = Im{y[n]} χ 2 k Chi-Squared distribution with k degrees of freedom f T H0 (x) PDF of the Test Statistic under H 0 xi

12 Notation Description Γ( ) Gamma function Γ(a, b) Gamma distribution with shape parameter a and rate parameter b N (µ, σ 2 ) Normal distribution with mean µ and variance σ 2 P rx Received/sensed power at the SU U Uniform distribution χ 2 k (λ) Noncentral Chi-Squared distribution with k degrees of freedom and noncentrality parameter λ f T H1 (x) PDF of the Test Statistic under H 1 Q(, ) Lower Regularized Gamma function P (, ) Upper Regularized Gamma function d d-parameter, difference between the means of the PDF of the Test Statistic under both hypothesis f T H0 (x) Normal approximated PDF of the Test statistic under H 0 Q( ) Q-funtion Q M (, ) Marcum Q-function of order M f T H1 (x) Normal approximated PDF of the Test statistic under H 1 σ h Scale parameter of the fading gain h 2 γ Average SNR for fading channels L Number of implemented receiver antennas at the SU (SLC case) κ l Weighting factor for the Test Statistic in receiver l (SLC case) T SLC (y 1,..., y L ) Test Statistic (SLC case) = L α l P l=1 σw 2 Vector with the L fading gains γ ΣL h 2 L xii

13 Dedicat a la meva família.

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15 Chapter 1 Introduction The demand for data transmission using wireless communications is considerably increasing day by day. Moreover, almost all the spectrum bands, which are appropriate for mobile communications, are occupied and already used by existing communications. Therefore, new techniques, for instance Dynamic Spectrum Access (DSA), have been required to solve this problem. Cognitive Radio Networks [1] are meant to be a viable solution for this issue, increasing the efficiency of the spectrum usage. Considering a certain B under license, two different types of users can be defined: Primary Users (PUs): These users hold the license of the spectrum band. Secondary Users (SUs): These users try to exploit side information about their environment in order to improve spectrum utilization. This side information is related to the activity of the primary users with which the secondary users share the spectrum. SUs seek to use the same band as the PUs do, without excessively impacting them. There might be secondary licenses for the SUs. In this deployment a Cognitive Relay (CR) device is considered, which is a network element that works as a SU, exploiting the underutilized band B resources. Its aim is to provide coverage to indoor devices through dynamically accessing the spectrum [2]. Therefore, the CR, which belongs to the Mobile Network Operator (MNO), has an indoor antenna and an outdoor antenna. Figure 1.1 shows a scenario having a CR. Although the CR has the system intelligence, it is connected with a SU through a Backhaul link in order to obtain a network-id. Besides, there are three different Cognitive Radio Network paradigms: Underlay, Overlay and Interweave paradigm [3]. In each of them, the SU has a different role. Underlay System: The SUs are allowed to operate although the PU is using the spectrum band. However, either an interference threshold or a primary user performance degradation bound is set. A tipical technique is to spread the SU signal over a wide bandwidth so that the Power Spectral Density (PSD) of the SUs signal remains below the noise floor [4]. Another option is to use low power transmissions. Since the interference constraints are really restrictive, Underlay Systems might only be used in specific cases, such as short range communications between SUs. 1

16 Overlay System: In this paradigm the SUs are also allowed to transmit while the PU is using the spectrum band. Nevertheless, the major difference is that the SUs have an a priori knowledge of the PUs transmitted data sequence and its encoding. This information can be exploited in the SUs transmitters with sophisticated signal processing techniques, in order to simultaneously improve the performance of primary users and transmit data. The SU transmitter usually amplifies the primary user signal to accomplish a certain Signal-to- Interference-plus-Noise Ratio (SINR). Interweave System: These systems are based on the idea that there are some temporary space-time-frequency voids, often referred as spectrum holes [5], in which no data is being transmitted by the PU. Hence, the resources of the spectrum band can be further exploited by the SUs. The spectrum band can only be used when the licensed user is not using it, i.e. only an opportunistic usage is allowed. Such a system periodically monitors the radio spectrum in order to detect spectrum holes. Figure 1.1: Scenario of a typical Cognitive Radio Network illustrating an interweave paradigm [2]. To sum up, while Overlay and Underlay Systems allow concurrent primary and secondary transmissions, Interweave Systems, which are the focus of this thesis, are not meant for simultaneous transmission. In Interweave Systems, to ensure that the selected spectrum band B is available for secondary usage, spectrum sensing techniques are required. Figure 1.2 shows the different techniques. Considering the non-cooperative systems, three different spectrum sensing techniques can be found [6]. Matched filter detection: The SU has a priori knowledge of PU signal. The matched filter can be seen as a correlation of the PU signal and a reference signal. Furthermore, the output value of the filter is higher when the PU signal is present. Cyclostationary feature detection: This technique, if the PU is present, exploits the periodicity in the received signal. It is robust to noise uncertainties and performs better than energy detection for lower Signal-to-Noise Ratio (SNR) scenarios [7]. 2

17 Spectrum Sensing Matched Filter Detection Cyclostationary Feature Detection Energy Detection Figure 1.2: Classification of the spectrum sensing techniques [8] Energy detection: Does not require a priori knowledge of the PU signal, since it is based on the sensed energy. Thus, higher energy values are expected when the PU is present and lower values when it is absent. All of these techniques can be seen as binary classifiers: PU signal SP Classifier decision: present/absent Figure 1.3: Block diagram of the sensing process The block SP stands for Signal Processing, which is explained in Chapter 2. This thesis is motivated by the CR s capacity to make an opportunistic usage of the spectrum by using an Energy Detector (ED). Energy detection technique is widely extended, due to its simplicity and no requirement on a priori knowledge of the PU signal. An ED estimates the presence of the PU signal by computing the average energy received and comparing it with a given threshold λ. Thus, energy detection can be reduced to a binary problem represented by an hypothesis test: { w[n] if H 0 y[n] = h (1.1) αx[n] + w[n] if H 1 where: H 0 : Hypothesis for absence of PU. H 1 : Hypothesis for presence of PU. y[n]: Processed received/sensed signal at the SU. x[n]: Processed PU received signal.each sample is assumed to be a Random Variable (RV). w[n]: Processed Channel AWGN noise. Each sample is assumed to be a RV. h: It is a RV that denotes the fading coefficient. α: It is a deterministic value that stands for the Path-Loss attenuation. 3

18 As it is indicated in next sections, only a certain channel of bandwidth B is considered. Additionally, all signals are assumed to be sampled in accordance to Nyquist criterion to give a discrete-time model. The PU presence estimation is done by averaging the energy over N samples of the processed received signal: T (y) = 1 N N y[n] 2 (1.2) Since every sample y[n] is a RV, T (y) is also a RV, which is referred as the Test Statistic. n=1 The block diagram of an ED is shown in Figure 1.4. y[n] y(t) SP 2 1/N N n=1 T (y) = 1/N N n=1 y[n] 2 y[n] 2 Buffer N Figure 1.4: Block diagram of an ED where the buffer stores the samples y[n] n {1,..., N}. The decision rule consists on comparing the energy value T (y) with the threshold λ. This threshold can be set according to a fixed probability of false alarm value P fa or a fixed probability of detection P d. P fa = Prob(T (y) > λ H 0 ) λ = f(p fa, T (y)) P d = Prob(T (y) > λ H 1 ) λ = f(p d, T (y)) (1.3) The probability of false alarm P fa is defined as the probability of deciding that the PU is using the channel when it is actually not being used (i.e. false positive). In addition, the probability of detection P d is defined as the probability of deciding that the PU is using the channel when it is being used (i.e. true positive). Both maximizing the probability of detection and minimizing the probability of false alarm imply improving the detector s performance. However, in terms of Cognitive Radio Network it is more important to maximize P d, in order to avoid interfering the PU. To correctly decide the threshold λ, the Probability Density Function (PDF) of T (y) has to be known. In the literature there are already some theoretical characterisations of its density function, which, however, are only valid for specific transmitted signals [7, 9, 10]. Since in this thesis another transmitted signal has been considered, a new probability characterisation of the Test Statistic RV has to be done. 4

19 The purpose of this thesis is to validate, through hardware, the diversity improvement of an ED s performance in different scenarios. At first, the system setup that is used the project deployment is described in Chapter 2. In Chapter 3 a single antenna detector is considered whereas in Chapter 4 an implementation of a Square-Law Combiner (SLC) in the detector is done. Finally, Chapter 5 summarizes the conclusions of this work and suggests future research directions. 5

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21 Chapter 2 System Setup In this system deployment there are three main elementes: Primary Transmitter (PT) Cognitive Relay, i.e. Secondary Transmitter (CR/ST) Channel Figure 2.1 shows the scenario. x(t) y(t) PT Channel CR/ST {H 0, H 1 } Figure 2.1: Block diagram of the channel Phase-Shift Keying (PSK) is widely used in existing technologies due to its simplicity. For instance, the wireless LAN standard, IEEE b-1999 [11], and the higher-speed wireless LAN standard, IEEE g-2003 [12], use a variety of different PSK modulations depending on the data rate required. Due to their extended usage in practical environments, it is a good idea to consider the received PU signal x[n] to be PSK modulated. However, in order to reduce the complexity of the problem a complex sinusoid is considered to be the received PU signal. Note that an important characteristic of PSK modulated signals is that their power remains constant in the discrete domain, just as complex sinusoids do. Complex sinusoids, can be seen as M-PSK modulated signals, being M a high number. Thus, an IQ-demodulator is needed at the receiver. The received PU signal is supposed to be already demodulated. x(t) = P e j2πfct (2.1) where f c refers to the carrier frequency. The PU signal is band-limited with E [x(t)] = 0 and Var [x(t)] = P the transmitted power. Figure 2.2 shows a complex sinusoid along time. 7

22 Figure 2.2: Complex sinusoid [13] As a preliminary setup, AWGN channel is considered. This depicts a Line-Of-Sight (LOS) condition, i.e. the transmitted signal travels over the air directly without passing through any obstruction, and the Path-Loss effect, due to free-space propagation, are considered. For this initial approximation, an USRP B210 has the role of the ST and a signal generator is used as the PT. Moreover, when considering the Non-Line-Of-Sight (NLOS) condition, where the transmitted signal may be reflected, refracted, diffracted, absorbed or scattered, a fading channel model is used. In this case, two USRPs are used for both the PT and the ST. 1 AWGN Channel The signal generator Rohde & Schwarz SMU200A has been used to model the primary user signal transmission. Figure 2.3 shows the block diagram of the transmitter, displayed on the screen of the signal generator. Note that f cent = GHz is the center frequency, whereas f c = 55 khz is the carrier frequency. An ISM band has been chosen, since they are license-free and meant for amateur purposes. 8

23 Figure 2.3: Rohde & Schwarz model SMU200A Signal Generator For spectrum sensing an USRP B210 has been deployed, cf. Figure 2.4. This device is connected to a computer that executes GNU Radio Companion. The USRP is responsible of detecting and storing the received signal from the channel, i.e. the sensed PU signal, in a buffer. In this case, the buffer is the memory of the laptop. Figure 2.4: USRP B210 deployment hardware Figure 2.5 shows the structure of the block SP seen in Figure 1.3 and Figure 1.4. The following operations are held: 9

24 Figure 2.5: Receiver GNU Radio block diagram when a SLC is implemented at the SU(used in AWGN channel and Fading channel) Receiving and sampling the signal: This is done by the UHD: USRP Source block using a given center frequency value and a given sampling frequency. In our system the sampling frequency is f s = 150 khz. The received, discretized and demodulated PU signal can now be reformulated: x RF [n] = P e j2πfcn where F c = f c /f s is the discrete carrier frequency Filtering the received signal: The Band Pass Filter block filters the received signal, reducing its bandwidth to B f = 30 khz, in order to increase the SNR at the SU γ. Mixing down the signal: Mixing down prevents from aliasing to appear when decimation is done. This is done by multiplying the filtered received signal by a cosine, which is generated with the Signal Source block, at the carrier frequency f c. Decimation: This is done by the Keep M in N block to avoid oversampling, i.e. correlation between signal samples. The ratio is 5 to 1 (i.e. M = 5), since f s /B f = 5. After this, the processed receiver signal y[n] is obtained, where x[n] = P e j2πfɛn is the received, processed and discretized received PU signal and F ɛ is the residual discrete frequency after processing the received signal x(t). Ideally F ɛ = 0 Hz, that is x(t) should be a DC-signal. Nevertheless, due to local oscillator errors and channel effects the received signal is not perfectly mixed down. These errors can occur due to temperature dependency, ageing of the hardware, Doppler shift etc [14, 15]. 10

25 2 Fading Channel For this case, as stated before, two USRPs are used both in reception (CR/ST) and transmission (PT). The reason behind it is that the power of the received PU signal is not constant, since the fading coefficient h changes over time according to the coherence time. The signal was generated in MATLAB and can be expressed as: S/N coh i=1 h i p Ncoh [n kn coh ] x RF [n] where S is the total number of generated samples, N coh is the discrete coherence time, h i denotes the i-th fading coefficient and p N [n n 0 ] is the discrete rectangular pulse defined as: p N [n n 0 ] = { 1 if n 0 < n < n 0 + N 1 0 if otherwise Figure 2.6 shows the PT block diagram, which only consists of two blocks. The File Source block loads the MATLAB generated signal and the UHD: USRP Sink block sends the signal. Figure 2.6: Transmitter GNU Radio block diagram (used in fading channel) Figure 2.7 shows the block diagram of the receiver (CR/ST). It is exactly the same as one branch of the block diagram in Figure 2.5. Figure 2.7: Receiver GNU Radio block diagram (Used in AWGN and a Fading channel) Figure 2.8 shows the amplitude change in the processed received signal y[n] due to fading effects. 11

26 Figure 2.8: Received signal in a fading channel The parameter N should be fixed in order to correctly compare the performance in the different scenarios. The choice of this parameter is really relevant for fading scenarios, where the fading coefficient is assumed to stay constant within the coherence time T coh. To find out an approximation for the coherence interval N coh for these scenarios, a wireless signal was captured with the USRP while walking inside a room. It was important not to leave the room, since the effect of shadowing (i.e. long-term fading) could appear. Afterwards the autocorrelation of its energy samples was computed. The 60% discrete coherence time can then be defined as (2.2) [16]. N coh60% = 1 2 [ ( R y 2( n) ) ( R y 2( n)) arg n>0 max = 0.6 arg R y 2(0) n<0 min = 0.6) ] (2.2) R y 2(0) where R y 2( n) is the time correlation function of the energy samples of the processed received signal y[n] 2 in the discrete domain. The continuous value can be obtained as T coh N coh M/f s = 129 ms. Figure 2.9 shows that N coh60% 3808 samples. Coherence time estimation 0.95 Autocorrelation of the energy samples signal Coherence time threshold 0.6 Normalized autocorrelation X: 3808 Y: sample Figure 2.9: Autocorrelation of the energy samples of the processed received signal y[n] with the threshold set by the coherence time, 0.6 It makes sense to choose N N coh, since not all the coherence time is used to sense but also to transmit data, as shown in the Figure In this thesis N = 100 is considered, 12

27 Figure 2.10: USRP s activity along the coherence time T coh 13

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29 Chapter 3 Single Antenna Initially, the SU is deployed with a single antenna. The results are easily extendible to a multiple antenna deployment. At first an AWGN channel is considered and the fading effects will be incorporated subsequently. 1 AWGN Channel In this case the channel consists on AWGN. PT x(t) {H 0, H 1 } α + y(t) ST/CR w(t) AWGN Channel Figure 3.1: Block diagram of AWGN channel with a single antenna a the SU The received signal y[n] can be expressed as (3.1), depending on the hypothesis. { w[n] if H 0 y[n] = (3.1) αx[n] + w[n] if H1 Where w[n] stands for an AWGN signal, i.e. w[n] CN (0, σ 2 w). Assuming that both the real and the imaginary parts of w[n] are i.i.d., the variances of both are equal to σ2 w 2. Signals x[n] and w[n] are considered to be uncorrelated. Moreover, α denotes the Path-Loss. In order to correctly decide if there is a signal or not, the average energy of the received signal samples is computed. Next, the performance of the energy detector is characterized by terms of the PDF of the Test Statistic T (y) and the Receiver Operating Characteristic (ROC) curve. 1.1 Hypothesis 0 Since the PU is absent, the received signal is y[n] = w[n] and the Test Statistic is given by (3.2). 15

30 T (y) H 0 = 1 N N y[n] 2 = 1 N n=1 = T (y R ) + T (y I ) N w[n] 2 = 1 N n=1 N w R [n] N n=1 N w I [n] 2 n=1 (3.2) where w R [n] and w I [n] stand for Re { w[n] } and Im { w[n] } respectively. T (y R ) and T (y I ) are supposed to be independent and identically distributed. Note that y R [n] = w R [n] N (0, σ2 w 2 ) and y I [n] = w I [n] N (0, σ2 w 2 ) Probability Density Function of Test Statistic Let ŷ R [n] = y R[n] σ w 2 = w R[n] 2 σ w, then it is defined: T R (y R ) = N ŷ R [n] 2 = n=1 N ( ) yr [n] 2 N 2 σ w = 2 y σw 2 R [n] 2 = 2N T (y σw 2 R ) n=1 As a result of the sum of N i.i.d. standard Normal distributions, T (yr ) is Chi-Squared distributed with N degrees of freedom [17, pp. 38]. n=1 T (y R ) χ 2 N And equivalently T (y I ) χ 2 N, where T (y I ) = 2N σ 2 w T I (y I ). Then, T (y) = T (yr ) + T (y I ) follows also a Chi-Squared distribution with k = 2N degrees of freedom [18]: T (y) H 0 χ 2 2N Finally, the Test Statistic RV is expressed as T (y) H 0 = σ2 w 2N T (y) H 0, which has a scaled Chi- Squared distribution [19, pp. 24], i.e. a Gamma distribution with shape parameter N and rate parameter N/σw 2 [20]: T (y) H 0 Γ(N, N/σw). 2 f T H0 (x) = 2N σ 2 w ( ) 2N f T0 σw 2 x = (N/σw) 2 N Γ(N) x N 1 e N σ 2 x w if x 0 0 otherwise where f T H0 (x) denotes the PDF of T (y) H 0, f T0 (x) stands for the PDF of T (y) H 0 and Γ( ) refers to the Gamma function [20]. Normal Approximation For mathematical tractability, the distribution of T (y) H0 has been approximated by a Normal distribution according to the Central Limit Theorem (CLT). The higher is N, the more accurate is the approximation. Given a Gamma distributed RV X Γ(a, b), where a is the shape and b is the rate factor, its mean is given by E {X} = a/b and its variance by Var {X} = a/b 2 [21]. 16

31 Mean E { T (y) H 0 } = N N/σ 2 w = σ 2 w (3.3) Variance PDF Var { T (y) H 0 } = (N/σ 2 w ) 2 = σ4 w N T (y) ( ) H0 N σw, 2 σ4 w N (3.4) (3.5) 1.2 Hypothesis 1 Under H 1 the received signal is expressed as y[n] = αx[n] + w[n]. The Test Statistic is given by (3.6). T (y) H 1 = 1 N N y[n] 2 = 1 N n=1 N αx[n] + w[n] 2 (3.6) In addition, the received power at the SU is defined by (3.7) and the SNR at the SU is given by (3.8). n=1 P rx = αp + σ 2 w (3.7) γ = αp σ 2 w (3.8) Probability Density Function of y[n] Taking into consideration the expression of y[n] under H 1 : y[n] = αp e jθ[n] + w[n] (3.9) where θ[n] = 2πF ɛ n is the phase of the received PU signal. Note that θ[n] varies in time, i.e. θ[n] U[ π, π]. Considering a certain time point n (i.e. a specific value of θ 0 ), then the received signal consists of a circular bivariate Normal RV with non-zero mean. If the magnitude is applied, a Rician distributed RV is obtained. ( σ ) w y θ0 [n] Rician αp, 2 Which has (3.10) as the PDF [19, pp ]. ( ) x 2 + αp αp x x f y[n] (x; θ) = I 0 σw/2 2 σw/2 2 e σw 2 if x 0 0 otherwise (3.10) 17

32 where I 0 is the zero-order modified Bessel function of the first kind [22, pp. 3]. Since the PDF of y[n] does not depend on the angle value θ, the PDF for all n [1,..., N] is: f y[n] (x) = π π f y[n] (x; θ) f θ (θ)dθ = f y[n] (x; θ) Finally: y[n] Rician( αp, σw 2 ). Bear in mind, that as the SNR γ increases, it is possible to approximate a Rician distribution by a Normal distribution with µ y = αp and σ y = σw 2 [23] Probability Density Function of Test Statistic Given a RV X Rician(ν, σ), then (X/σ) 2 has a Noncentral Chi-Squared distribution with two degrees of freedom and noncentrality parameter (ν/σ) 2 [24]. Let ŷ[n] = y[n] 2 σ w, then ŷ[n] 2 χ 2 2 (2γ), that is the square of the modulus of the normalized received signal y[n] follows a Noncentral Chi-Squared distribution with k = 2 and noncentrality parameter λ = 2γ. Taking this into consideration the following expression is defined: T (y) H 1 = N ŷ[n] 2 = n=1 N ( ) y[n] 2 N 2 σ w = 2 y[n] 2 σw 2 n=1 Since T (y) H1 is the sum of N i.i.d. Noncentral Chi-Squared distributed RVs it is also Chi- Squared distributed [18]. T (y) H 1 χ 2 2N(2Nγ) Finally, the Test Statistic RV can be expressed as T (y) H1 = σ2 w 2N T (y) H1, which is a scaled Noncentral Chi-Squared distribution [19, pp. 26]: f T H1 (x) = 2N σ 2 w ( ) 2N f T H1 σw 2 x = N σw 2 ( e N σw 2 (x+αp ) x αp n=1 ) 1 2 (N 1) I N 1 ( 2N σ 2 w αp x ) if x 0 0 otherwise (3.11) where f T H1 (x) denotes the PDF of T (y) H 1, f T H1 (x) stands for the PDF of T (y) H 1 and I M ( ) is the modified Bessel function of order M [22, pp. 3]. Normal Approximation Considering a RV X χ 2 k (λ), then E {X} = k + λ and Var {X} = 2(k + 2λ) [17, pp ]. Mean E { T (y) H 1 } = E { σ 2 w 2N T (y) H 1 } = σ2 w 2N E { T (y) H1 } = σ 2 w 2N (2N + 2Nγ) = σ 2 w + αp = P rx (3.12) 18

33 Variance Var { T (y) H 1 } = Var { σ 2 w 2N T (y) H1 } = σ 4 w 4N 2 Var { T (y) H1 } = σ 4 w 4N 2 2(2N + 4Nγ) = σ4 w N (1 + 2γ) (3.13) If the noise variance is known, γ can be previously calculated: γ = αp σ 2 w = P rx σ 2 w 1 (3.14) PDF T (y) ( ) H1 N P rx, σ4 w N (1 + 2γ) (3.15) 1.3 Performance Analysis In this section the ROC for the detector is obtained and analysed. ROC shows the performance of the binary detector depending on the selected threshold value λ. First of all, the probabilities that characterize any detector have to be defined: P fa (λ) = Prob(T (y) > λ H 0 ): Probability of false alarm. This probability quantifies the probability of detecting that the PU is using the channel when it is not. P cr (λ) = Prob(T (y) < λ H 0 ): Probability of correct rejection. It measures the probability of correctly detecting that the PU is not using the channel. Take into account that both probabilities are complementary, i.e. P fa + P cr = 1. PDF H 0 H 1 d σ 2 λ αp + σ 2 T (y) Figure 3.2: False alarm probability (red) and probability of correct rejection (green) P d = Prob(T (y) > λ H 1 ): Probability of detection. This probability shows the capability of the SU to detect the PU signal when the primary user is using the channel. P md = Prob(T (y) < λ H 1 ): Probability of miss detection. It shows the probability of missing a PU signal detection. 19

34 Note, again, that P d + P md = 1. PDF H 0 H 1 d σ 2 λ αp + σ 2 T (y) Figure 3.3: Detection probability (green) and miss detection probability (red) Bear in mind, the d-parameter, which measures the distance between the mean received power for H 1 and the mean received power for H 0. If the Test Statistics under both hypothesis can be characterised by Normal PDF, the d-parameter is easy to compute. d = E {T (y) H 1 } E {T (y) H 0 } (3.16) Note that the larger is the number of samples N, the better are the approximations Probability of False Alarm To compute P fa only H 0 is relevant, i.e. T (y) H0. Reviewing the Cumulative Density Function (CDF) of a Gamma distributed RV and (3.3): P fa (λ) = Prob(T (y) > λ H 0 ) = ( = Q N, N ) σw 2 λ λ f T H0 (x)dx = 1 λ 0 ( f T H0 (x)dx = 1 P N, N ) σw 2 λ (3.17) where P (, ) and Q(, ) are the Lower and Upper Regularized Gamma function respectively [25]. Let f T H0 (x) be the Normal approximated PDF of the Test Statistic under H 0, then: P fa (λ) = Prob(T (y) > λ H 0 ) = λ ) λ σ 2 f T H0 (x)dx = Q( w σw/ 2 N (3.18) where Q( x µ σ ) = x f x(x)dx if f x (x) is the PDF of X N (µ, σ 2 ) is the Q-function. 20

35 1.3.2 Probability of Detection To compute P d only the H 1 is relevant, i.e. T (y) H1. Contemplating (3.11): ( ) { 2N 2N P d (λ) = Prob(T (y) > λ H 1 ) = f T H1 (x)dx = λ λ σw 2 f T1 σw 2 x dx = z = 2N } σw 2 x ( ) 2Nγ, (3.19) 2N = f T1 (z)dz = Q N λ 2N σw 2 λ where Q M (, ) = is the Marcum Q-function of order M [26]. σ 2 w Let f T H1 (x) be the Normal approximated PDF of the Test Statistic under H 1, then: ( ) λ P rx P d (λ) = Prob(T (y) > λ H 1 ) = f T H1 (x)(x)dx = Q σ λ w 2 N 1 + 2γ Results In this last subsection the results obtained in the deployment are compared with the theoretical expressions obtained in previous sections, both for P fa and P d as also for the PDFs of the Test Statistic. A scenario with γ = 5 db and N = 100 samples, as stated in Chapter 2, is chosen. In this system, with a sampling rate of f s = 150 khz and a decimation factor M = 5, 100 energy samples are computed within 3. 3 ms. However, reducing the value of N results into a less accurate Normal approximation for the distribution of the Test Statistic. According to (3.16), the d-parameter is computed as following: d AW GN = P rx σ 2 w = αp Figure 3.4 shows the probability distributions of the Test Statistic under both hypothesis, both experimental and theoretical. Note that there is a substantial overlapping area. Figure 3.5 shows the theoretical and the experimental results for the ROC curve. These figures demonstrate that the Normal approximation is valid, since the calculus is much more easier and the results are pretty accurate though. 21

36 x 10 9 AWGN Channel (SNR = 5dB ; N = 100) PDF of the test statistic T(y) H0 Theoretical (Normal Approximation) H0 Experimental H1 Theoretical (Normal Approximation) H1 Experimental T(y) x Figure 3.4: PDFs for H 0 and H 1 both experimental and theoretical obtained for an AWGN channel 1 ROC Curve (SNR = 5dB ; N = 100) 0.9 Probability of Detection (Pd) Theoretical results (Normal approximation) [AWGN] Experimental results [AWGN] Probability of False Alarm (Pfa) Figure 3.5: ROC curve for an AWGN channel 22

37 2 Fading Channel Once the AWGN channel has been characterized, the effect of fading is now considered. This model consists of an instantaneous complex value h = h e jφ which is multiplied by the PU signal. Bear in mind, that this value is assumed to remain constant within the coherence time T coh (equivalent to N coh in discrete domain). The block diagram of this channel is shown in Figure 3.6. x(t) α PT + {H 0, H 1 } h w(t) y(t) ST/CR Fading AWGN Channel Figure 3.6: Block diagram of the fading channel with a single antenna at the SU Once the received signal has been correctly sampled, its expression, depending on the hypothesis, is given by: { w[n] if H 0 y[n] = h (3.20) αx[n] + w[n] if H 1 Under H 0 the received signal is exactly the same as in (3.1) and therefore the probability of false alarm remains the same. Nevertheless, under H 1 the coefficient h changes the PDF of the Test Statistic T (y) H 1, so that the expression of P d has to be reformulated. The distribution of the fading coefficient h can vary depending on the scenario. In this thesis only the case with Rayleigh fading is contemplated, which is a good model for tropospheric and ionospheric signal propagation and for urban environments effects on radio communications [27]. Let h CN (0, 2σh 2) be the fading coefficient, then its real part h R and its imaginary part h I are Normal distributed with zero mean and, assuming that both are i.i.d., variance σh 2. Therefore, the fading coefficient is h = h 2 R + h2 I Rayleigh(σ h). If h is Rayleigh distributed with scale factor σ h, then the fading gain h 2 is exponentially distributed with rate parameter λ = 1 [28] 2σh 2 with the PDF shown in (3.21) [29]. 1 e x 2σ f h 2(x) = 2σh 2 h 2 if x 0 (3.21) 0 otherwise Furthermore, the phase of h is uniformly distributed: φ h U[0, 2π]. It is important to note the fact that the SNR for this case is given by (3.22). γ( h 2 ) = h 2 αp σ 2 w (3.22) 23

38 Besides, an average SNR is defined by (3.23). { h 2 αp } γ = E σw 2 = 2σ2 h αp σw 2 (3.23) 2.1 Probability Density Function of Test Statistic For a given constant h value, it is known that the product h αx[n] is also deterministic with power h 2 αp and phase θ [n] = θ[n] + φ h = 2πF ɛ n + φ h, which remains uniformly distributed, i.e. θ [n] U[0, 2π]. Hence, for a given h value the scenario is the same as in the AWGN Channel, i.e. the Test Statistic also follows a scaled Noncentral Chi-Square distribution. f Rayleigh T H 1 (x; h 2 ) = N ) σw 2 e N σw 2 (x+ h 2 1 αp )( x 2 (N 1) ( 2N h 2 I N 1 αp σw 2 h ) αp x (3.24) where f Rayleigh T H 1 (x; h 2 ) denotes the PDF of T (y) H1 for a given h Rayleigh distributed coefficient. If all h 2 values are considered, which are exponentially distributed, (3.24) needs to be averaged over (3.21) in order to remove the uncertainty introduced by the fading coefficient: f Rayleigh T H 1 (x) = f Rayleigh T H 1 (x; h 2 ) f h 2( h 2 )d h 2 (3.25) Finally, the PDF of the Test Statistic under two hypothesis is expressed by (3.26). f Rayleigh T (x) = for x 0. (N/σw) 2 N Γ(N) 0 x N 1 e N σw 2 x N e N σw 2 σw 2 (x+ h 2 αp ) ( x h 2 αp ) 1 2 (N 1) I N 1 ( 2N σ 2 w h ) 1 αp x 2σh 2 if H 0 e h 2 2σ h 2 d h 2 if H 1 (3.26) However, the expression for H 1 is too complex to work with. Therefore, in this case, the Normal approximation is very useful. Normal Approximation When the PT is absent, the Test Statistic is exactly the same as in (3.5): T (y) H 0 N ( σ 2 w, σ 4 w/n ) Under H 1, a first approximation of (3.24) to a Normal distribution for a given h 2 value, according to (3.15), is given by: T (y; h 2 ) ( H 1 N P rx ( h 2 ), σ4 ( w 1 + 2γ( h 2 ) )) N 24

39 where the received power is now defined P rx ( h 2 ) = h 2 αp + σ 2 w. f Rayleigh Let T (x) denote the Normal approximated PDF of the Test Statistic for a Rayleigh fading channel. Following the same procedure as for (3.26), the following PDF is obtained for x 0: f Rayleigh T (x) = (x σ 2 w )2 e 2σw 4 /N if H 0 1 2πσ 2 w / N 1 0 2πσ 4 w /N(1+2γ( h 2 )) e (x Prx( h 2 )) 2 2σ 4 w /N(1+2γ( h 2 )) 1 2σ 2 h e h 2 (3.27) 2σ h 2 d h 2 if H 1 Expression (3.27) is only useful if αp is known. It is assumed that the noise variance, fading scale parameter and the average SNR are known at the SU. Thus, using expression (3.23) the attenuated transmitted power αp can be computed. 2.2 Performance Analysis Knowing the results for expressions of P fa and P d for an AWGN channel, it is easily extended to the Rayleigh channel case. Note that the probability of false alarm remains the same. P Rayleigh fa (λ) = Prob(T (y) > λ H 0 ) = λ ( f T H0 (x)dx = Q N, N ) σw 2 λ Given (3.25) and (3.26) the probability of detection is expressed as: (3.28) P Rayleigh d (λ) = Prob(T (y) > λ H 1 ) = f h 2( h 2 ) d h 2 dx = 1 2σ 2 h e h 2 ] 2σ h 2 d h 2 dx = ( 2N I N 1 σw 2 h αp x 1 2σh 2 e h 2 2σ h 2 d h 2 λ f Rayleigh T H 1 (x) dx = [ λ 0 0 ) ] dx N [ 1 λ 2σ 2 h σ 2 w λ e N σw 2 (x+ h 2 αp )( x h 2 αp N σ 2 w f Rayleigh T H 1 (x, h 2 ) ) 1 e N σw 2 (x+ h 2 αp )( x h 2 αp e h 2 2σ h 2 d h 2 = Appraising the Normal approximation, the probabilities are given by: 0 2 (N 1) I N 1 ( 2N ) 1 2 (N 1) Q N ( 2Nγ( h 2 ), σ 2 w ) 2N σw 2 λ h ) αp x (3.29) ) P Rayleigh λ σ 2 fa (λ) = Prob(T (y) > λ H 0 ) = f T H0 (x)dx = Q( w λ σw/ 2 N (3.30) and 25

40 P Rayleigh d (λ) = Prob(T (y) > λ H 1 ) = f Rayleigh T H 1 (x) dx = λ [ f h 2( h 2 ) d h 2 1 dx = 1 2σh 2 e h 2 2σ h 2 d h 2 = 0 0 Q ( λ λ 2πσ 4 w (1 + 2γ( h 2 ))/N e λ P rx ( h 2 ) σ 2 w N 1 + 2γ( h 2 ) ) 1 2σ 2 h f Rayleigh T H 1 (x, h 2 ) e h 2 2σ h 2 d h 2 (x Prx( h 2 )) 2 2σ 4 w /N (1+2γ( h 2 ) ) ] dx (3.31) Results As done in the Performance Analysis of an AWGN Channel, the experimental results and theoretical expressions are contrasted. For the theoretical expressions the Normal approximation is taken into consideration. Note, that the SNR γ does not remain constant over time, in so much that it changes according to the fading coefficient variations. Accordingly, the average SNR γ is now used as the reference value. The same scenario is chosen: γ = 5 db and N = 100 samples. x Rayleigh Fading Channel (SNR = 5dB ; N = 100) H0 Theoretical (Normal Approximation) H0 Experimental H1 Theoretical (Normal Approximation) H1 Experimental PDF of the test statistic T(y) T(y) x 10 9 Figure 3.7: Probability density functions for H 0 and H 1 both experimental and theoretical obtained for a Rayleigh fading channel 26

41 1 ROC Curve (SNR = 5dB ; N = 100) 0.9 Probability of Detection (Pd) Theoretical results (Normal approximation) [Rayleigh Fading] Experimental results [Rayleigh Fading] Probability of False Alarm (Pfa) Figure 3.8: ROC curve for a Rayleigh fading channel Finally, the theoretical expressions for the PDFs (Figure 3.7) and the ROC curve (Figure 3.8) have been validated through hardware. 27

42 28

43 Chapter 4 Square-Law Combiner Once the performance of the energy detector has been analysed with a single antenna at the receiver, the problem is extended to L antennas in order to verify the energy detection performance improvement. Hence, the expressions obtained in the previous chapter are fundamental for this chapter. Bear in mind, that although a generic case with L antennas at the SU is contemplated, L = 2 antennas have been implemented in the deployment. Following the same procedure as in Chapter 3, only additive white Gaussian noise is taken at first into consideration. 1 AWGN Channel The received signal for channel l is given by: y l [n] = { w l [n] if H 0 αl x[n] + w l [n] if H 1 (4.1) for l {1,..., L}. Where y l [n] denotes the received signal, α l stands for the Path-Loss and w l [n] CN (0, σ 2 w l ) stands for the noise for channel l respectively. All the noises are assumed to be uncorrelated. Considering L SU antennas, there are L different Test Statistics: T (y l ) = 1 N N y l [n] 2 (4.2) There are several ways to work with multiple Test Statistics, however in this thesis only the SLC technique is considered. In this scheme, the outputs of the Square-Law Devices (SLD) are linearly combined. Hence, a new decision statistic is defined as: 29 n=1

44 T SLC (y 1,..., y L ) = L κ l T (y l ) (4.3) where κ l stands for the weighting factor for channel l. The weighting factors are often assumed to be κ l = 1 l [1,..., L], which in the literature is referred as Equal Gain Combining (EGC). l=1 x(t) α1 + y 1 (t) SLD 1 T (y 1 ) PT {H 0, H 1 } x(t) w 1 (t) AWGN 1 y 2 (t) α2 + SLD 2 T (y 2 ) Combiner T SLC ( y 1, y 2 ) w 2 (t) AWGN 2 Figure 4.1: Block diagram of the channel with SLC for L = 2 Analogically to Chapter 3, the Test Statistic can also be approximated by a Normal distribution. 1.1 Hypothesis 0 Considering L Test Statistic RVs { T (y 1 ) H 0,..., T (y L ) H 0 }, all of them independent and identically Gamma-distributed, the exact expression of the distribution of their weighted sum is really complex. Therefore, approximations might be used. The interested reader can address to [20, 30, 31]. Examining the EGC case, i.e. κ l = κ l {1,..., L}, and equal noise variances σw 2 l = σw 2 l {1,..., L}, it is proofed that T SLC (y 1,..., y L ) H 0 = L l=1 κt (y l) Γ ( ) N LN, κσ [21], being w 2 (4.4) its PDF. Normal Approximation ( ) LN N κσw f TSLC H 0 (x) = 2 Γ(LN) x LN 1 e N κσ 2 x w if x 0 0 otherwise The extension from a single antenna to L antennas with a generic set of weighting factors {κ 1,..., κ L } is direct. Mean E { T SLC (y 1,..., y L ) } H L 0 = κ l E {T (y l ) H 0 } = l=1 30 L κ l σw 2 l (4.4) l=1

45 Variance Var { T SLC (y 1,..., y L ) H 0 } = L l=1 κ 2 l Var { T (y l ) H 0 } = 1 N L κ 2 l σ4 w l (4.5) l=1 PDF T SLC (y 1,..., y L ) ( L H 0 N κ l σw 2 l, 1 N l=1 ) L κ 2 l σ4 w l l=1 (4.6) 1.2 Hypothesis 1 Under this hypothesis the Test Statistic is given by: T (y l ) = 1 N N y[n] 2 = 1 N n=1 N α l x[n] + w l [n] 2 Moreover, the received power for channel l is defined by (4.7) and the SNR for channel l is given by (4.8). n= Probability Density Function of Test Statistic P rxl = α l P + σ 2 w l (4.7) γ l = α lp σ 2 w l (4.8) The mathematical development of the exact PDF of the Test Statistic for H 1 is, again, very complex. Therefore, only the exact expression for the EGC case with equal noise variances is obtained. The interested reader might address to [32, 33]. T (y l ) H 1 = N ŷ l [n] 2 = n=1 N ( ) yl [n] 2 N 2 σ w = 2 σw 2 n=1 n=1 y l [n] 2 = 2N σw 2 T (y l ) H 1 Note that T (y l ) H 1 χ 2 2N (2Nγ l) is Noncentral Chi-Squared distributed. Knowing that T (y l ) H 1 = σw 2 T 2N (y l ) H 1, the Test Statistic expression is reformulated as: T SLC (y 1,..., y L ) H 1 = L l=1 κt (y l ) H 1 = κ σ2 w 2N L l=1 T (y l ) H 1 = κ σ2 w 2N T SLC (y 1,..., y L ) H 1 Since T SLC (y 1,..., y L ) H 1 is the sum of L i.i.d. Noncentral Chi-Squared distributed RVs it is also Noncentral Chi-Squared distributed, i.e. TSLC (y 1,..., y L ) H 1 χ 2 2LN (2Nγ Σ L ), where γ ΣL = L l=1 γ l = L l=1 [18]. α l P σ 2 w Finally, the PDF of T SLC (y 1,..., y L ) H 1, with the assumptions made, is a scaled Noncentral Chi-Squared distribution for x 0: 31

46 f TSLC H 1 (x) = 2N ( ) 2N κσw 2 f TSLC H 1 κσw 2 x = N κσw 2 e N σw 2 ( x k + L l=1 α lp ) ( x κ L l=1 α lp ) 1 2 (LN 1) I LN 1 (2N σ 2 w L ) l=1 α (4.9) lp x k where f TSLC H 1 is the PDF of T SLC (y 1,..., y L ) H 1 and f TSLC H1 is the PDF of T SLC (y 1,..., y L ) H 1 for equal variances and EGC. Bear in mind, that the values of α l P are usually unknown at the receiver. However, they can estimated with the received power P rxl and the noise floor estimation σ 2 w l for each channel, similarly as in (3.14). Normal Approximation The extension from a single antenna to L antennas with a generic set of weighting factors {κ 1,..., κ L } is direct. Mean E { T SLC (y 1,..., y L ) } H L 1 = κ l E { T (y l ) } H L 1 = κ l P rxl (4.10) l=1 l=1 Variance Var { T SLC (y 1,..., y L ) H1 } = L l=1 κ 2 l Var { T SLC (y l ) H 1 } = 1 N L κ 2 l σ4 w l (1 + 2γ l ) (4.11) l=1 PDF T SLC (y 1,..., y L ) ( L H1 N κ l P rxl, 1 N l=1 L κ 2 ( ) ) l σ4 w l 1 + 2γl (4.12) l=1 1.3 Performance Analysis Using the results from a single antenna defined in Chapter 3, the new expressions of P fa and P d for a certain threshold λ are easily found. The EGC case with equal noise variances is considered for the exact expressions. Knowing the CDF of a Gamma distributed RV, the probability of false alarm is defined as: ) P faslc (λ) = Prob (T SLC (y 1,..., y L ) > λ H 0 = λ ( f TSLC H 0 (x)dx = Q LN, N κσ 2 w ) λ (4.13) Considering the CDF of a Noncentral Chi-Squared distributed RV, the detection probability is defined as: 32

47 ) P dslc (λ) = Prob (T SLC (y 1,..., y L ) > λ H 1 = = { z = 2N } ( 2NγΣL κσw 2 x = f 2N TSLC H 1 (z)dz = Q LN, κσw 2 λ λ ( 2N 2N f TSLC H 1 (x)dx = λ κσw 2 f TSLC H 1 ) 2N λ κσ 2 w κσ 2 w ) x dx (4.14) Taking into consideration the Normal approximations (4.6) and (4.12) the probabilities, for a generic set of L weighting factors, are reformulated as: ) P faslc (λ) = Prob (T SLC (y 1,..., y L ) > λ H 0 = ) λ L l=1 = Q( κ lσw 2 l 1/N L l=1 κ2 l σ4 w l λ f TSLC H 0 (x)dx (4.15) ( P dslc (λ) = Prob T SLC (y 1,..., y L ) > λ H 1 ) = ( λ ) L l=1 = Q κ lp rxl 1/N L ( ) l=1 κ2 l σ4 w l 1 + 2γl λ f TSLC H 1 (x)dx (4.16) Results Analogically to the previous chapter, empirical and experimental results for P faslc and P dslc are plotted together for L = 2 antennas and κ 1 = κ 2 = 1. The same scenario is chosen, that is γ = 5 db and N = 100 samples. Figure 4.3 shows the experimental and the analytical Normal approximation PDFs under both hypothesis, which fit quite good together once again. Besides, the implementation of the SLC scheme has improved the ROC curve, that is for a given threshold value λ the P fa has decreased and the P d has increased. Consequently, the detection process is more accurate and therefore the primary user suffers less interference. This is mainly due to the fact that more samples are received. In addition, the d-parameter has substantially increased: d SLC AW GN = κ 1 P rx1 + κ 2 P rx2 (k 1 σ 2 w 1 + k 2 σ 2 w 2 ) = κ 1 α 1 P + κ 2 α 2 P Assuming that α 1 = α 2 = α and κ 1 = κ 2 = 1, the d-gain factor can be defined as: gaw SLC GN = dslc AW GN d AW GN = 2 33

48 x 10 9 AWGN Channel SLC (SNR = 5dB ; N = 100) 4 H0 Theoretical (Normal Approximation) H0 Experimental H1 Theoretical (Normal Approximation) H1 Experimental PDF of the test statistic T(y) T(y) x 10 9 Figure 4.2: PDF of the Test Statistic under H 0 and H 1 both experimental and theoretical obtained for an AWGN channel with SLC 1 ROC Curve (SNR = 5dB ; N = 100) Probability of Detection (Pd) Theoretical results (Normal approximation) [AWGN] Experimental results [AWGN] Theoretical results (Normal approximation) [AWGN + SLC] Experimental results [AWGN + SLC] Probability of False Alarm (Pfa) Figure 4.3: ROC curves 34

49 2 Fading Channel This chapter considers all the different aspects mentioned in previous chapters: AWGN noise Fading Multiple antennas in reception (deployment with L = 2 antennas) Therefore, the system can now be referred as a diversity system. The received signal in channel l has now the following expression: y l [n] = { w l [n] if H 0 h l αl x[n] + w l [n] if H 1 (4.17) where h l stands for the fading complex coefficient of channel l. The fading is considered to be Rayleigh, therefore h l Rayleigh(σ hl ) for each channel l. The square of the fading coefficient is then exponentially distributed, as seen in Chapter 3. h l 2 f hl 2( h l 2 1 2σ ) = e 2σh 2 h 2 l ifx 0 l 0 otherwise (4.18) In addition, the received power for channel l is given by (4.19) and the SNR for channel l is defined by (4.20). P rxl ( h l 2 ) = h l 2 α l P (4.19) Furthermore, the average SNR for channel l is given by (4.21). γ( h l 2 ) = h l 2 α l P σ 2 w l (4.20) γ l = E { hl 2 A 2 l σ 2 w l The block diagram of this scenario for L = 2 is shown by Figure (4.4). 35 } = 2σ2 h l α l P σ 2 w l (4.21)

50 Fading 1 x(t) h 1 y 1 (t) T (y 1 ) α1 SLD 1 + PT {H 0, H 1 } x(t) α2 w 1 (t) AWGN 1 + y 2 (t) SLD 2 T (y 2 ) Combiner T SLC ( y 1, y 2 ) h 2 Fading 2 w 2 (t) AWGN 2 Figure 4.4: Block diagram of the Rayleigh fading channel with SLC for L = 2 The Test Statistic is computed in the same way as in the AWGN Channel with SLC, defined in (4.2). 2.1 Probability Density Function of Test Statistic For the exact expressions (i.e. Noncentral Chi-Squared) the EGC case with equal noise variances is reviewed again. On the one hand, when the PU is absent, i.e. H 0, the Test Statistic remains the same and is given by (4.4). On the other hand, regarding (4.9) and considering the coefficients h l 2 to be deterministic for all l {1,..., L}, the PDF for the Test Statistic under H 1 for x 0 is given by (4.22). f Rayleigh T SLC H 1 (x; h L 2 ) = N ( κσw 2 e N σw 2 ( x κ + L l=1 h l 2 α l P ) x κ L l=1 h l 2 α l P ( L 2N l=1 I h l 2 α l P LN 1 x κ σ 2 w ) 1 2 (LN 1) ) (4.22) where h L 2 = [ h h L 2 ] is the vector containing fading coefficients for the L channels. Considering the PDFs, i.e. the randomness, of the fading coefficients the Test Statistic T SLC ( y1,..., y L ) H1 is defined as: 36