The received signal of a SU can be expressed by two hypothesis model as [7] [8] ,

y ( n ) = { v ( n )             H 0 s ( n ) + v ( n )   H 1 (1)

where, n = 0 , 1 , 2 , ⋯ , N − 1 are the sampling instants, v(n) is awgn at receiving end and s(n) is the transmitted signal by a PU. Here the received signal vector y follows Gaussian pdf like, N ( 0 , σ 2 I ) under H₀ and N ( μ , σ 2 I ) under H₁.

Applying Neyman-Pearson (NP) detection rule,

N ( μ , σ 2 I ) N ( 0 , σ 2 I ) > γ

⇒ 1 ( 2 π σ 2 ) N / 2 e − 1 2 σ 2 ∑ n = 0 N − 1 { y ( n ) − A } 2 1 ( 2 π σ 2 ) N / 2 e − 1 2 σ 2 ∑ n = 0 N − 1 y 2 ( n ) > γ

Solving we get the test statistics,

1 N ∑ n = 0 N − 1 y ( n ) = T ( y ) = σ 2 N A ln ( γ ) + A 2 = g (2)

It can be shown that the test statistics T(y) follows,

T ( y ) = { N ( 0 , σ 2 / N )     H 0 N ( A , σ 2 / N )       H 1 (3)

The pdf of T,

f T ( t ) = 1 2π exp { − t 2 / ( 2 σ 2 / N ) } = 1 2π exp { − N t 2 / 2 σ 2 } (4)

The probability of false alarm (under Hypothesis H₀)

P F A = P ( T > g ) = 1 − P ( T ≤ g ) = 1 − F T (g)

where F_T(g) is the cdf of T. Now the cdf of T can be expressd as,

F T ( g ) = ∫ − ∞ g 1 2π exp { − N t 2 / 2 σ 2 } d t = 1 − Q ( g σ 2 / N ) ;

where Q ( x ) 1 2π ∫ x ∞ e − t 2 / 2 d t

Therefore the probability of false alarm,

P F A = Q ( g σ 2 / N ) (5)

The probability of detection (under Hypothesis H₁),

P D = P ( T > g ) = 1 − P ( T ≤ g ) = 1 − ∫ − ∞ g 1 2π exp { − N ( t − A ) 2 / 2 σ 2 } d t = 1 − Q ( A − g σ 2 / N ) = Q ( g − A σ 2 / N ) (6)

Combining Equation (5) and Equation (6),

P D = Q ( Q − 1 ( P F A ) − N A 2 σ 2 ) (7)

In fusion center (FC) three different types of decision is taken for proper detection as explained in [9] [10] . If the FC takes the decision using AND logic then,

P D _ F C = P D N and P F A _ F C = P F A N (8)

for OR logic case,

P D _ F C = 1 − ( 1 − P D ) N and P F A _ F C = 1 − ( 1 − P F A ) N (9)

and for majority decision case,

P D _ F C = ∑ j = ⌈ N 2 ⌉ N ( N j ) P D j ( 1 − P D ) N − j and P F A _ F C = ∑ j = ⌈ N 2 ⌉ N ( N j ) P F A j ( 1 − P F A ) N − j (10)

Recent papers deal with primary or secondary user emulation attack based on Equation (8) to Equation (10) to eliminate the malicious user at FC.

Let the nth sample of received signal is x(n) and the mean of such N samples,

T = 1 N ∑ n = 0 N − 1 x ( n ) is taken as a random variable under normal distribution, N(μ,

σ²/N). We have to determine the probability of the random variable T exceeds a threshold value γ. The steps of Monte Carlo simulation are shown below:

Algorithm

1) Generate N Gaussian random variable with mean μ = 0 and variance σ² using, x = s q r t ( v a r ) ∗ r a n d n ( N , 1 ) .

2) Compute, T = 1 N ∑ n = 0 N − 1 x ( n ) .

3) Repeat step 1 and 2 to generate, T = [ T ( 1 ) T ( 2 ) T ( 3 ) ⋯ T ( M ) ] .

4) Count the number of element of the array T exceeds the threshold γ. If this number is k then the probability P ( T > γ ) = k / M .

5) In co-operative spectrum sensing technique, each CR sense the signal and sends its decision in FC (fusion center) to take the final decision based on k-out-of-N voting rule of [11] or the rules of [9] [10] .

The theoretical value, P ( T > γ ) = Q ( γ σ 2 / N ) = 0.5 e r f c ( γ 2 σ 2 / N ) . In a

CRN, if the threshold SNR at receiving end is taken as, γ = τ then P(T > τ) = k/M provides the probability of false alarm under Hypothesis H₀ and 1−P(T > τ) under H₁ provides the probability of detection.

First of all let us simulate the probability density function (pdf) of Rayleigh distributed random variable. Let x is a random variable follows Rayleigh pdf like,

f X ( x ) = x σ 2 e − x 2 / 2 σ 2 ; where σ 2 is the variance of x. (11)

The cdf of x will be,

∴ F X ( x ) = 1 2 σ 2 ∫ 0 x 2 e − y / 2 σ 2 d y = 1 − e − x 2 / 2 σ 2 (12)

Now solving for the r.v. x,

e − x 2 / 2 σ 2 = 1 − F X ( x ) ⇒ x = σ ( − 2 ln { 1 − F X ( x ) } ) (13)

Here the cdf F X ( x ) is actually a probability can be treated as a random number over [0 1].

In wireless communication the Rayleigh pdf of instantaneous SNR, γ is expressed as [12] [13] ,

f Γ ( γ ) = 1 γ a v e − γ γ a v ; where γ a v is the average SNR (14)

The cdf of γ ,

F Γ ( γ ) = 1 − e − γ γ a v (15)

Now solving for the r.v. γ ,

1 − e − γ γ a v = F Γ ( γ ) ∴ γ = − γ a v ln ( 1 − F Γ ( γ ) ) (16)

The cdf, F Γ ( γ ) is now a random number in the range of [0 1], to generate the SNR, γ follows Rayleigh pdf.

Let us now train a neural network with simulated false alarm, detection and SNR; where the theoretical false alarm and detection is taken as the target value. In this section first we use backpropagation algorithm then Hopfield algorithm. In back propagation algorithm (BP) the signal flows in forward direction i.e. from input to output through hidden layers. The output and the desired signal are compared and error is detected. The weighting vector at the input terminals of output nodes is changed using steepest descent algorithm to reduce the amount of error. The weight vector of hidden layers is updated using same algorithm from the expression of error at those layers. The process is continued till the input of first hidden layer i.e. error reduction operation (or update of weight) flows in backward direction. The above procedure is iterated till error at output falls below the threshold.

For simplicity of understanding let us consider an ANN of three layers. The steps of BP are given bases on [14] [15] .

1) Apply the vector X p = [ x p 1   x p 2   x p 3   ⋯   x p N ] at pth instant as the input sequence;

2) The output of the jth node of hidden layer,

i p j = f h ( ∑ j = 1 L w j i h x p i + θ j h ) ;

where f h ( x ) is the activation function of hidden layer;

3) The final output of kth node,

o p k = f o ( ∑ j = 1 L w k j o i p j + θ k o ) ;

where f o ( x ) is the activation function of output;

4) Calculate error at output and hidden layer as,

δ p k o = ( y p k − o p k ) f ′ k o ( ∑ j = 1 L w k j o i p j + θ k o ) and δ p j h = f ′ k h ( ∑ j = 1 L w j i h x p i + θ j h ) ∑ k δ p k o w k j o ;

5) The weight is updated on output and hidden layer as,

w k j o ( t + 1 ) = w k j o ( t ) + η δ p k o i p j and w j i h ( t + 1 ) = w j i h ( t ) + η δ p j h x i ;

when error becomes less than a threshold for the training vector then algorithm stops. The algorithm next selects another input vector and its desired vector.

In this paper we also use Hopfield model to merge simulation and theoretical results. In Hopfield model the processing unit has two outputs: inverting and non-inverting. Each output is connected back to the all inputs except itself. If w_ij is the weight between ith input and jth output then the output at ith output before the non-linear function f(.) is:

S i = ∑ j n w i j O j + x i − Φ i ; for i = 1 , 2 , 3 , ⋯ , n ;

where O_j is the output of jth neuron after the nonlinear function f(.), x_i is the external input at node i, and Φ_i is the threshold of ith neuron. The Hopfield algorithm flows the steps like [15] [16] :

1) Initialize weights w_ij = +1 or −1.

2) The input x_i and output O_i are initialized as: x_i = O_i(k) = +1 or −1; i = 1 , 2 , 3 , ⋯ , n ; the threshold, Φ_i = 0 and k = 0.

3) Evaluate S i = ∑ j n w i j O j + x i − Φ i and O i ( k ) = f ( ∑ j n w i j O j + x i − Φ i ) ; where k = 1.

4) Evaluate O i ( k + 1 ) = f ( ∑ j n w i j O i ( k ) ) ; 0 ≤ j ≤ N − 1.

5) Repeat step 4 until output remains unchanged.

The next section provides results based on both algorithms of this section.