2.1. Matched Filter Detection

When the information of the primary user signal is well-known to the cognitive radio, the matched filter is the optimal linear filter for maximizing the signal to noise ratio (SNR) in the presence of additive Gaussian noise (Sahai at el., 2004, Akyildiz at el., 2006). In cognitive radio system, the matched filter is obtained by correlating a known signal, or primary user signal template, with an unknown signal to detect the presence of primary user signal in the unknown signal. This is equivalent to convolving the unknown signal with a time-reversed version of the template. According to the hypothesis model for transmitter detection, the received signal x(n) can be expressed as

where s(n) and sc(n) are original signal and pilot signal transmitted by primary user, which are orthogonal. η is the ratio of power allocated to pilot signal. The correlation function of received signal x(n) and unit vector of pilot signal s^c(n) is given by

The Equation (3) can also be expressed as

where P is the power of the pilot signal, and a is noise variance. Then the decision function can be given by

where the threshold y is determined by the probability of false alarm P_FA. The probability of false alarm P_FA and detection P_D are given as follows

where Q(·) is the generalized Marcum Q-function, which can be defined by

Combining Equation (6) and Equation (7), we can obtain N by eliminating y .

According to the Equation (9), O (1/ SNR) samples are required to meet a probability of error constraint, thus the main advantage of the matched filter is that it requires less time to achieve high processing gain. However, as a coherent detection method, matched filter detection requires a priori knowledge of the primary user signal such as the modulation type and order, the pulse shape, and the packet format. Although most of the priori knowledge can be obtained from pilot, preambles, synchronization word or spreading codes of the primary user network systems, an obvious shortcoming is that the cognitive radio user requires specific receivers for different types of primary user signal.

2.2. Energy Detection

If the receiver cannot gather sufficient information about the primary user signal, for example, if the power of the random Gaussian noise is only known to the receiver, the optimal detector is an energy detector (Akyildiz at el., 2006, Digham at el., 2003). In order to measure the energy of the received signal, the output signal of bandpass filter with bandwidth W is squared and integrated over the observation interval T. Finally, the output of the integrator, x(t), is compared with a threshold, y, to decide whether a licensed user is

Fig. 2. The principal of energy detection.

Since it is easy to implement, the recent work on detection of the primary user has generally adopted the energy detection. Although this method can be implemented without any prior knowledge of the primar₂y user signal, it also has some drawbacks. Since energy detection is non-coherent, O(1/SNR ) samples are required to meet a probability of error constraint. Moreover, the threshold selection for energy detection is highly susceptible to uncertainty in background noise and interference, and it can only determine the presence of the signal without differentiating signal types. However, the largest advantage of energy detection is simple and low complexity.

2.3. Cyclostationary Feature Detection

The cyclostationary feature is intentionally embedded in the physical properties of a communication signal, which may be easily generated, manipulated, detected and analyzed using low complexity transceiver architectures. This feature is present in all transmitted signals, requires little signalling overhead and may be detected using short signal observation times, and thus it can be used for primary user signal detection and recognition. Recent research efforts exploit the cyclostationary features of signals via spectral correlation analysis as a method for spectrum sensing (Digham at el., 2003, Sahai at el., 2004, Ghasemi & Sousa, 2005), which has been found to be superior to simple energy detection and matched filtering. Energy detection can only detect whether or not a signal in present and utilizing a matched filter system requires extensive knowledge about the channel and signals that are to be identified. The method, which is not susceptible to in-band interference, can be used to detect and classify different types of signal. Conventional signal classification approaches are mainly based on decision theory (Polydoros & Kim, 1990, Sapiano & Martin, 1996, Sills,

1999, Wei & Mendel, 2000, Hang at el., 2001) and statistical pattern recognition (Nandi &

Azzouz, 1995, Azzouz & Nandi, 1995, Azzouz & Nandi, 1996, Nandi & Azzouz, 1998). In the work by Hang (Hang at el., 2001), a binary decision tree is designed to signal classification. However, it's difficult to obtain the decision thresholds and rules, which needs a large amount of calculation. For more efficient and reliable performance, a novel approach based on multilayer linear perceptron network for signal classification in cognitive radio is studied by Fehske (Fehske at el., 2005). Support vector machine (SVM) is a new statistical pattern recognition approach, which is based on structural risk minimization principle (Vapnik, 1995). Compared with the conversional methods based on empirical risk minimization like artificial neural network (ANN), it has been found to give better generalization and better performance for small training examples. In the next section, a novel approach of signal classification for cognitive radios combining the cyclostationary features and SVM is proposed.

3.1. System Framework

As our signal classification scheme combining spectral correlation analysis and SVM is based on statistical pattern recognition, which mainly consists of three modules (Han, 2003): feature extraction, classifier design and classification decision. Feature extraction is typically the first stage in any classification system in general, and in our spectrum sensing systems in particular. Given signal set to be classified, the feature parameters of different classes of signal and rules for classifier should be determined first. In order to achieve better classification performance, selected feature parameter should be insensitive with the SNR variation, and then a proper classifier is designed for specific classification problem using training data with known signal types. When the error probability of the classifier achieves a specific threshold, the classifier can be used for signal classification and recognition. In our scheme, there are three procedures adopted for primary user signal recognition:

1) Pro-processing procedure:

Several feature parameters are extracted via spectral correlation analysis first. The feature parameters insensitive with the SNR variation are selected as feature vector x1,x2,⋯,xM . After that, the feature vector and signal type yk are used to form the training set (x1,x2,⋯,xM,yk) for classifier.

2)Training and learning procedure:

The SVM classifier is trained using selected feature parameters in the training set. By utilizing a nonlinear SVM, an amount of calculation for training is performed offline, thus the computational complexity is reduced. The optimal classification plane for SVM is obtained in this procedure via training and learning.

3)Test procedure

Selected feature parameters extracted for received signals are inputted well-trained SVM classifier for primary user signal detection and recognition.

The framework of our scheme combining spectral correlation analysis and SVM is shown in Fig. 3.

Figure 2.

System framework.

Although SVM is a better choice for the classifier, the selection of feature parameters has direct impact on the performance of the classification algorithm. In the next section, we will discuss the first step, how to choose spectral coherence characteristic parameters for our scheme.

3.2. Spectral Correlation Analysis

Many signals used in communication systems exhibit periodicities of their second order statistical parameters due to the operations such as sampling, modulating, multiplexing and coding. These cyclostationary properties, which are named as spectral correlation features, can be used for signal detection and recognition (Gardner, 1987).

In order to analyze the cyclostationary features of the signal x(t), two key functions are typically utilized. The cyclic autocorrelation function (CAF) is used for time domain analysis, which can be expressed as

The spectral correlation function (SCF), which exhibits the spectral correlation of the signal x(t), is obtained from the Fourier transform of the cyclic autocorrelation in Equation (12) (Gardner & Franks, 1975).

Where α is the cycle frequency, f is the spectral frequency and SxTα(t,f) is the cyclic periodogram of Sxα(f)

where the Fourier transform of the function x(u) on the bounded time interval [t-T/2,t + T/2] is definedas

The correlation coefficient for the SCF between frequency components f±α/2 , which is known as spectral coherence coefficient (SCC), can be calculated by

The magnitude of the SCC ranges from 0 to 1 with a = 0 for all f. Different signal classes (i.e. AM, ASK, FSK, PSK, MSK, QPSK) can be distinguished based on several characteristic parameters of SCF and SCC.

In practical situations, however, the number of observation samples at the sensor is limited. Therefore, the spectral correlation function needs to be estimated from a finite set of samples. In general, two methods are used for spectral correlation estimation including time-domain averaging and frequency-domain smoothing (Gardner & Spooner, 1988). In this section, the frequency-smoothing method is used for spectral correlation estimation, which can be expressed as follows.

where FS=1/(N-1)TS T_S is the frequency increment, TS is the cyclic sampling interval and the length of the sample is N=Δt/TS . Thus, the data-tapering window WΔt(kTS) is of the width Δf=M·FS . The Spectral correlation functions of some typical signals can be obtained by frequency-smoothing method, which are shown in Fig. 4 (Gurman at el., 2008).

Figure 3.

Even visually in the above figures, the spectral correlation functions of the different modulation types possess distinct characteristics. It is this fact that allows the successful application of the pattern recognition techniques to achieve primary user signal detection and recognition. In order to obtain better robustness of the proposed algorithm, some features less sensitive with SNR should be chosen for the classifier. Assumed that the received signal s = s + n, where s, n are the transmitted signal and additional white Gaussian noise. The feature parameter of the received signal has a better classification where performance which is insensitive with SNR variation, if it satisfies

Based on the calculation of the spectral correlation function, we can obtain the spectral correlation magnitude surface of different types of signal. According to above analysis, several spectral correlation features can be extracted for distinction of different modulation types. Typically, four key features x1,x2,x3,x4 which are sensitive with modulation types and insensitive with SNR variation are listed in Table. 1.

Table.1.

Typical value of spectral correlation features. Four key features x₁,x₂,x₃,x₄are described as follows.

•x₁: Number of 5 pulse on f - domain of SCF

Let a = 0 in Equation (17), the SCF is transformed into S"(f), thus x₁ can be obtained from the ichnography of S _x(f).

•x₂: Number of cyclic spectral line on a -domain of SCF

Let f = 0 in Equation (17), the SCF is transformed into S (0), and x₂ can be obtained from the ichnography of S (0).

•x₃: Average energy of cyclic spectral line on a - domain of SCF

The Average energy of cyclic spectral line on a - domain of SCF can be computed by the equation as follows:

• x₄: Maximum value of SCC

The spectral coherence coefficient can be obtained via Equation (16) and Equation (17). Then, the maximum value of SCC is computed as key feature x₄.

In the simulation experiments, 4000 features are extracted from the signal for every trial. In order to prevent numerical computational errors, the features need to be normalized by subtracting mean of each feature from the original feature and dividing the result by the standard deviation of the same feature.

After normalization, the feature vector x=(x′1,x′2,x′3,x′4) can be generated as the input of the signal classifier.

3.3. Support Vector Machine

The traditional statistical theory is primarily based on the asymptotic principle, which provides conclusion only for the situation where the sample size is tending to infinity. However, in most practical applications, the samples are usually limited so that it is difficult to achieve the desired results via existing methods. Statistical Learning Theory is a novel statistical theory based on small sample statistics by Vapnik (Vapnik, 1995). Compared to the conventional statistical theory, statistical learning theory mainly concerns the statistic principles when samples are limited, especially the properties of learning procedure in such cases. Statistical learning theory provides us a new framework for the general learning problem, which not only considers the asymptotic performance but obtains the optimal results under the condition of limited information. In order to study the generalization performance and the speed of uniform convergence, a series of indicators used to evaluate the learning performance of function sets are defined in statistical learning theory. One of the most important concepts is Vapnik-Chervonenkis(VC) dimension which was originally defined by Vladimir Vapnik and Alexey Chervonenkis in 1971. VC-dimension is a measure of learning machine complexity or the capacity of a statistical learning algorithm, which is the cardinality of the largest set of points that the algorithm can shatter. The learning machine is more complex with a greater VC-dimension. Statistical learning theory provides a novel strategy that balances the empirical risk and confidence interval. A nested subset sequence is chosen from the given set of functions according to the size of the VC dimension. For a given subset, the minimal value of the empirical risk can be obtained as the minimal true risk, which is illustrated in Fig. 5. This method is named as Structural risk minimization (SRM) which was also coined by Vapnik and Chervonekis in 1974. The principle of SRM is to provide a method to reach the trade-off between hypothesis space complexity (the VC dimension of approximating functions) and the quality of fitting the training data (Wang, 2007). The procedure is described in detail as follows.

Assumed a function set Q(x,α),α∈Λ has a set S, which has a structure shown in Fig. 5. The structure is defined by a nested subset sequence S1,S2,⋯,Sn , which satisfies

The elements of the above structure have two properties as follows.

(1) The VC dimension of each subset h_k is limited and satisfies

(2) Any element in the structure S_k contains a set of totally bounded function

or contains a function set which satisfies the following inequality for some (p, t _k)

For a given set of observation set z1,z2,⋯,zl , SRM aims to choose the proper function Q(z,αlk) who makes the empirical risk minimal in subset Sk with smallest guaranteed risk. After performing empirical risk minimization on each subset, the subset whose sum of empirical risk and VC confidence is selected. The optimal function Q(z,αlk) that makes the empirical risk minimal in selected subset is the optimal solution.

Figure 5.

The principle of Structural Risk Minimization.

Statistical learning theory also gives the required conditions for reasonable structure of function subset and the convergence property of actual risk in SRM principle. The actual risk is the sum of empirical risk and confidence interval. As the index of the elements in the structure increases, the empirical risk will be reduced with extended confidence interval. The smallest upper bound of the actual risk can be derived from a certain element in the structure. Support vector machine is a novel universal learning machine, which is widely used in the fields of pattern recognition, regression estimation and probability density. It is based on VC-dimension theory and SRM principle, which has a better generalization performance by reaching a trade-off between model complexity with limited sample data and capacity of the learning algorithm. The support vector machine was coined by Vapnik in the late 1960s on the foundation of statistical learning theory. It was originally developed for binary classification problem. The optimal solution of SVM for a linearly separable case was introduced by Vapnik. Later this was extended to non-separable cases. In the previous research, a common solution to classification problem of communication signals is artificial neural network (ANN), such as multilayer linear perceptron network. After the first preliminary studies, SVM have shown a remarkable efficiency, especially when compared with traditional artificial neural networks. The main advantage of SVM, with respect to ANN, consists in the structure of the learning algorithm, characterized by the resolution of a constrained quadratic programming problem (CQP), where the drawback of local minima is completely avoided (Boser at el., 1992, Cortes & Vapnik, 1995 , Scholkopf, 1995). Since the classification of communication signals is obvious to be a linearly non-separable problem, we will only discuss the computation of this optimization problem in this chapter. Given linearly non-separable classification problem, we suppose a training set is {(x_i, y_i )}, where x∈RN,y∈{+1,−1},i=1,...,1 . Assumed that the signal group can be classified by a hyperplane which is defined by

where the vector w defines the boundary of different classes of data, b is a scalar threshold and Φ(x)=‖x‖2/2 .

In order to solve this non-separable problem, the non-minus slack variables ξi are introduced, and then the training vectors must satisfy

The hyperplane, which makes O(w) = ||w|| /2 to be minimum, is named as the optimal hyperplane. All of the training vectors are correctly classified by it and the vectors of each class are separated with a maximum margin (Burges, 1997).

Usually, structuring a hyperplane is solved as a quadratic optimization problem that can be formulated as

where C is the penalty parameter, which is used to control the training error rate by different values.

Using a Lagrange multiplier technique, the optimization problem can be converted into

where _ai, (_i> 0 are Lagrange multiplier factors.

Given linearly non-separable classification problem, we can map the input data into a high dimensional feature space through some non-linear transformation which makes the data linearly separable (Devroye, 1996). Noted the above solution to linearly non-separable classification problem, only the inner product operation of the training samples is involved in the decision function. While structuring high dimensional feature space, the algorithm only use the inner product O (x_i )«O (x) in the space without separated O (x) or O (x_i). If we can find a function K satisfying Mercer condition (Daniel & James, 2000), which can be denoted as

where K(x_i, x) is the kernel function, which is utilized for mapping the input data to higher dimensional space in order to reduce the computational load. There are different kernel functions like polynomial, sigmoid and radial basis function (RBF) used in SVM, which are defined as follows.

1. Polynomial Kernel

A k-order polynomial classifier can be defined by Equation (30).

2. Radial Basis Function (RBF) Kernel

The width of the RBF kernel parameter o can be determined in general by an iterative process selecting an optimum value based on the full feature set. The main difference between RBF classifier and traditional RBF method is that each basis function in the RBF classifier corresponds to a support vector, which is automatically identified by the algorithm where the drawback of local minima is completely avoided.

3. Sigmoid Kernel

This kernel uses sigmoid function as inner product, which is equivalent to a multilayer perceptron with only one hidden layer. The number of node in hidden layer is automatically determined by algorithm.

Till now, the choice of the kernel functions was often used empirically, and this also became a theoretical drawback of SVM. A proper kernel function for a specific problem is dependent on the specific training sample data. In the practical applications, how to choose the proper model according to training sample set with better generalization ability is currently a research direction in the field of SVM. For a signal classification problem using cyclostationary features, we use an improved method of model selection based on kernel alignment, which will be described in Section 4.1 in detail. The choice of the kernel functions is studied via computer simulations and optimal results are achieved using radial-basis function (RBF) kernel function. A typical classification experiment using RBF kernel function based SVM is illustrated as follows

Figure 6.

A typical classification experiment using RBF kernel function (Sherrod, 2008).

After choosing the best kernel function, the dual representation of the optimization problem can be obtained by computing the derivatives with respect to w, b, ξi , which is described as

The resulting decision function is obtained as follows:

The architecture of the SVM classifier combining spectral correlation analysis is shown in Fig. 7.

4.2. Classifier Design

In order to compare the performance of different classifiers, two approaches based on existing methods, such as decision theory and artificial neural network, are introduced with spectral correlation features as training data.

4.2.2. Multilayer Linear Perceptron Network

Artificial Neural networks have long been considered for pattern recognition and modulation classification and have proven to be robust to a variety of conditions such as interfering signals and noise. In order to compare the classifier performance of artificial neural network and support vector machine, a signal classification approach using spectral correlation and neural networks, which was proposed by A. Fehske (Fehske at el., 2005), is introduced below. Due to its simplicity, a multilayer linear perceptron network (MLPN) with 4 neurons in the hidden layer was used for each signal class, and each input layer uses the normalized spectral correlation feature vector x' = (x'_t, x2, x'₃, x'₄) as input. Each MLPN was trained with a back propagation algorithm (Gupta, 2003) with an initial learning rate n = 0.05 decreasing with each epoch, a momentum constant a = 0.7, and an activation function tanh(x). The output of each MLPN is a continuous value in the range (-1, 1). The MAXNET structure shown in Fig. 10 simply chooses the signal whose MLPN outputs the largest value. A typical gradient descent algorithm can be used to solve the linearly non-separable signal classification problem, which can achieve minimal mean square error of expected output and actual output. The training results of all the MLPN are inputted into a simple MAXNET for final decision. The decision function of the MAXNET is defined by

z(k)=argmax[yi(k)]

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The signal classification approach using spectral correlation and MLPN is shown in Fig. 10.

Figure 10.

Architecture of the MLPN classifier based on spectral correlation analysis.