SNR Wall Reduction

The material in this Chapter is largely taken from [17].

When we use the energy detector, a significant problem is that it is suffers from an SNR wall when the noise power uncertainty is present [5],[23]. Caused by the noise uncertainty, the SNR wall is defined as an SNR threshold below which energy detection is absolutely impossible no matter how many samples are used. Now, we consider that there exists xdB uncertainty in noise power estimation, and then the actual noise power may take any value within ( ²

2

, _n

n ασ

α

σ ), where 10¹⁰

x

α = and σ_n²

is the estimated noise power. If the primary signal power is smaller then (

α ασ_n² −σⁿ² ),

then the energy detection will always fail. In other words, the SNR wall of energy detection is defined as follows:

where γ_w is SNR wall. Here we assume that the channels of the cognitive radio users experience block fading, and the block length is long enough so that errorless detection can be guaranteed if only the instantaneous SNR is greater then the SNR wall (γ ). While the number of the secondary users increases, the probability that the

1) ( log 10 ) ( log

10 _{10 10}

α α

γ = −

= _w

SNRwall ^(2.24)

instantaneous SNR on one of these users is greater then SNR wall increases. Once this probability exceeds the target overall probability of detection of the cognitive radio network, energy detection will work well. Therefore, cooperation equivalently decreases the SNR wall with a certain target probability of detection. Now, we will derive the equivalently SNR wall reduction achieved by cooperation among independently cognitive radio users.

Let γ_M be the minimum average SNR that meets the target overall probability of detection (P_{d_TAR}), when M independent secondary users are cooperating. In other

words, γ_M is equivalent the SNR wall of a M-secondary users network with the target overall probability of detection (P_{d_TAR}). Miss detection will happen if and only if the instantaneous SNR of all cognitive radio users are below γ_w, so we can obtained as follows:

Here we use the Nakagami channel [24]. In this case, the CDF of the instantaneous SNR (γ ) is obtained as follows:

where m is the Nakagami parameter, and ^{P m x =} _{Γ m}

∫

0^{xe−ttm−dt} 1

) ( ) 1 ,

( is the

normalized lower incomplete gamma function, and Γ(m) is the gamma function.

M w TAR

d TAR

m P

P _{_} =1− _{_} =(Pr(γ <γ )) ^(2.25)

) , ( ) Pr(

M w w

m m

P γ

γ γ

γ < = ^(2.26)

Now, we can obtain the equivalent SNR wall of a M-secondary users cooperative network as

where )P⁻¹(m,y is the inverse normalized lower incomplete function. Finally, we can obtain the equivalent SNR wall reduction relative to the single user as follows:

According to (2.28), we observe that the SNR wall reduction increases with the number of secondary user ( M ), independent of γ_w. For fixed m and P_m,TAR,

which means that the equivalent SNR wall of the cognitive radio network can be reduced to any arbitrarily low level as long as a sufficient number of the cooperating secondary users. Therefore, we use this result to improve the detection performance in chapter 3, the simulation can be shown that as the number of the cooperating secondary users increase, the detection performance becomes better.

)

Chapter 3 Distributed Cooperative Spectrum Sensing for

Two Cases

In this chapter, we propose an optimal linear cooperative structure for spectrum sensing in order to accurately detect the primary signal. In this structure, spectrum sensing is based on the linear combination of local statistics from individual cognitive radio, and we control the combining weights to combat the effect of channel fading.

Our objective is to minimize the interference to the primary user while the secondary users access the licensed band. So we optimize the modified deflection coefficient at the fusion center in order to improve the detection performance.

3.1 System Model

We consider a cognitive radio networks with M secondary users. The binary hypothesis test for spectrum sensing at the k-th time instant is expressed as follows:

where )s(k is the signal transmitted by the primary user and y_i(k) is the received signal by the i -th secondary user. The channel gain, h between each _i secondary user and the target primary user, is assumed to be fixed during a detection interval, and v_i(k)denotes the zero-mean additive white Gaussian noise (AWGN), i.e.

M

) be independent of each other.

As illustrated in Fig. 3-1, we use the energy detection, since it doesn’t require any a priori knowledge of primary signals and has much lower complexity then other detectors. Each secondary user computes its summary statistic u over a detection _i interval of 2n samples. i.e.

1 2

The summary statistic {u } are then transmitted to the fusion center through a _i fading channel and are corrupted by the zero-mean additive white Gaussian noise (AWGN), we can express as follows:

or

where the channel gain {g } between secondary users and fusion center are _i additive white Gaussian noise with zero-mean and variance σ , and they are _g²

i

assumed to be fixed during a detection interval, and the channel noise {n } are also _i additive white Gaussian noise with zero-mean and variance σ_n² ,

i.e.g_i ~ N(0,σ_g²),n_i ~ N(0,σ_n²). Finally, the fusion center computes the global test statistics,r as in (3.18), from the outputs {_c r }of the individual secondary users in a _i linear combination manner, and then r is used to make a global decision. _c

Fig. 3-1 A schematic representation of weighting cooperation for spectrum sensing in cognitive radio networks.

3.2 Cooperative Spectrum Sensing

In this section, we propose a optimal strategy for cooperative spectrum sensing.

Because we do not know the prior knowledge of the primary signals (i.e. the secondary user has limited information of the primary signals), the energy detection is optimal and the simplest, so we adopt energy detection as the local sensing rule,

which will be discussed as follows.

3.2.1 Local Sensing

We first consider local spectrum sensing at individual secondary users, and then we find out the local test statistics at each node. For the sequence of 2n samples over each detection interval, we define

which denotes the transmitted signal energy. The local test statistics of the i -th secondary user using energy detector are expressed as follows:

Since u is the sum of the squares of 2_i n Gaussian random variables, so we can show that u_i/σ_v² is a central chi-square χ² distribution with 2n degrees of freedom if H is true; otherwise, the ₀ u_i/σ_v² would be a non-central chi-square χ2 (η_i) distribution with 2n degrees of freedom and η_i is a non- centrality parameter. We can express as follows:

∑

⁻

=

=^{2 1}

0

)2

(

n k

s s k

E (3.5)

1 2 2

0

)

∑

⁻ (

=

= ⁿ

k i

i y k

u i=1,2,...,M (3.6)

where

is the local SNR (signal to noise ratio) at i -th secondary user. According to CLT (central limit theorem), if the number of samples is large enough, the test statistics u _i can be asymptotically normally distributed with mean

and variance

We can express simply as follows:

⎪⎩

for 2n is large enough. Now, for a single-CR spectrum sensing scheme, the decision rule at each secondary user is given by

where γ_i is the corresponding decision threshold. Therefore, secondary user i will have the probabilities of detection and false alarm, and we can express as the following Q-function:

and

Q : Calculates the tail probability of the zero mean unit variance Gaussian (.)

random variable, i.e. , ^{= +∞}

∫

⁻

In the cognitive radio networks, a larger probability of detection results in less interference to the primary users and a smaller probability of false alarm results in

0

higher spectrum efficiency. This is because based on the assumption that if no primary signals are detected, the secondary users use the channel (such that interference is generated in case of miss-detection); if a primary signals is detected (possibly a false alarm), the secondary users are restrained to use the channel (such that spectrum is wasted in case of false alarm).

3.2.2 Global Detection

As illustrated in Fig. 3-1, we transmit the local test statistic {u } to the fusion _i center via a channel and the zero-mean additive white Gaussian noise (AWGN),and then are multiplied by weights in a linear combination manner. Now, we consider the channel with two conditions. First, for simplification, we assume that the channel can be treated as constant AWGN channels which channel gains are constant one

(i.e.g_i =1). Second, we further consider that the channels are fading channels which channel gains are generated according to a normal distribution and assumed to be fixed during a detection interval.

I. Constant AWGN Cannel between Secondary User and Fusion Center

From (3.3) or (3.4), we assume that the channel gains are constant one (i.e.g_i =1), then we can express as follows:

i i

i u n

r = + i=1,2,...,M (3.15)

According to (3.11), since u_i ~ N(E[u_i],Var(u_i)), )n_i ~ N(0,σ_n² , so the received statistics {r } are normally distributed with mean _i

and variance

Once the fusion center receives {r }, a global test statistic _i r is calculated _c linearly as follows:

where the weight vector w=(w₁,w₂,...,w_M)^T satisfies ² 1

2 =

w and

⋅ 2 is the Euclidean norm. The weight vector is used to control the global spectrum detector.

The combining weight for the signal from a particular user represents its contribution to the global decision. For example, if a CR generates a high-SNR signal that may lead to correct detection on its own, it should be assigned a larger weight coefficient.

For the secondary users passing deep fading or shadowing, their weights are decreased in order to reduce their effect to the decision fusion. r =(r₁,r₂,...,r_M)^T is

the received vector. Since the received statistics {r } are Gaussian random variables, _i so their linear combination is also Gaussian. Then, r is normally distributed with _c mean

where

1

is a column vector that are all ones, and η =(η₁,η₂,...,η_M)^T is the SNR vector ,and variance

Therefore, the variances for different hypothesis are given by

where I denotes the identity matrix, and diag(.) is square diagonal matrix with the elements of a given vector on the diagonal. Therefore, we can express simply as follows:

Finally, to make decision on the presence of the primary signal, the global test statistics r is compared with a threshold _c T . _c

And then, the probabilities of detection and false alarm at the fusion center can be expressed as

generated according to a normal distribution and assumed to be fixed during a detection interval.

II. Fading Cannel between Secondary User and Fusion Center

From (3.3) or (3.4), the channel gain {g } between secondary users and fusion _i center are additive white Gaussian noise with zero-mean and variance σ_g², and they are assumed to be fixed during a detection interval, and the channel noise {n } are _i also additive white Gaussian noise with zero-mean and variance σ_n² ,

i.e. g_i ~ N(0,σ_g²),n_i ~ N(0,σ_n²) . And u_i ~N(E[u_i],Var(u_i)) . Without loss of generality, we assume that {g }, {_i n }, and {_i u } are independent of each other. _i Therefore, at the fusion center receives {r }, a global test statistic _i r is calculated _c linearly as follows:

Once again, according to CLT (central limit theorem), if the number of secondary users ( M ) is large enough, the global test statisticsr , can be asymptotically normally _c distributed with mean

∑

∑

= =

+

=

=

= ^M

i

i i i i M

i

T i i

c wr w r w gu n

r

1 1

)

( (3.27)

where {g } are assumed to be fixed during detection interval, and _i n_i ~ N(0,σ_n²) , u can be asymptotically normally distributed _i

So we can derive the mean as follows:

where g=(g₁,g₂,...,g_M)^T, _{M M} ^T

g (η₁g₁,η₂g₂,...,η g )

η = .

And variance

where K =E[(r−E[r])(r−E[r])^T] is covariance matrix.

Therefore, we can find the element of the covariance matrix for different hypothesis as follows:

w

and

However, we can observe that the covariance matrix is diagonal matrix which non-diagonal element are all zeros. Then, we can derive the variance

whereg² =(g₁²,g₂²,...,g_M² )^T,g²(n+η)=[g₁²(n+η₁),g₂²(n+η₂),...,g_M² (n+η_M)]^T. Finally, we can express that the global test statistics r are asymptotically normally _c distributed when M is large enough.

Then, to make decision on the presence of the primary signal, the global test statistics r is compared with a threshold _c T . _c

The probabilities of detection and false alarm at the fusion center can be expressed as

and

We see that the sensing performance of the linear detector depends largely on the weighting coefficient and the decision threshold. We next show how to design the optimal weight vector w in order to maximize the modified deflection coefficient.

0

3.3 Performance Optimization

For cognitive radio networks, the probabilities of detection and false alarm have unique relationship. Specifically, 1-P represents the probability of interference _d^(c) from secondary users on the primary users. On the other hand, P_f^(c) determines the

upper bound on the spectrum efficiency, where a large P_f^(c) usually results in low spectrum utilization. This is based on a typical assumption that if primary signals are detected, the secondary users do not use the licensed band, and if no primary signals are detected, the secondary users use the licensed band. In this section, we maximize the modified deflection coefficient in order to improve the detection performance.

From the mean and variance of r , we observe that the weight vector _c w plays an important role in controlling the PDF of the global test statistics r . To measure the _c effect of the PDF on the detection performance, we define a modified deflection coefficient as follows:

Now, we would like to maximize d under the unit norm constraint to find the _m² optimization of weight vector, i.e.

1 0 1

), (

) ], [ ], [

( ²

2

H c

H c H c

m Var r

r E r

d E −

=

(3.41)

1

subject to

) ( maximize

2 2 2

w = w d_m

(3.42)

Where ⋅₂ denotes the Euclidean norm. And then we consider two cases to find the weight coefficient. First, we consider that the channels are the constant AWGN channel, and we can obtain the follows from (3.41):

We solve the problem as follows. Since we have 4σ_v⁴[nI+diag(η)]+σ_n²I f0, so we can know its square root can be expressed as

Where is a diagonal matrix, Applying the linear transformation q=Dw gives

Where )λ_max(⋅ denotes the maximum eigenvalue of the matrix. Note that (a) follows the Rayleigh Ritz inequality and the equality is achieved if q=q^o, which is the eigenvector of the positive definite matrix D⁻¹ηη^TD⁻¹ corresponding to the

maximum eigenvalue. Therefore, we find the optimal solution of (3.42) is

which maximizes the modified deflection coefficient.

Now, we further consider that the channel are fading channel which channel gains are generated according to a normal distribution and assumed to be fixed during a detection interval. With the same step, we would like to maximize d under the unit _m² norm constraint to find the optimization of weight vector, and then we can obtain the follows from (3.41)

We solve the problem as the same, since we have 4σ_v⁴[diag(g²(n+η))]+σ_n²I f0, so we can also know its square root can be expressed as

2

Applying the linear transformation q=Dw gives

Where λ_max(⋅) denotes the maximum eigenvalue of the matrix. Note that (a) follows the Rayleigh Ritz inequality and the equality is achieved if q=q^o, which is the eigenvector of the positive definite matrix D^{−1 T}D⁻¹

g gη

η corresponding to the

maximum eigenvalue. Therefore, we find the optimal solution of (3.42) as the same step is

which maximizes the modified deflection coefficient. we can prove by the simulation results below, a larger value of d leads to a larger probability of _m² detection.

)

3.4 Simulation Result

In this section, the proposed approach is simulated numerically and compare with some other existing approaches. Firstly, we consider three or ten secondary users (M=3 or M=10) in the cognitive radio networks, and the secondary users sense the frequency spectrum independently. The channel gain between each secondary user and the target primary user is generated by a complex normal distribution (i.e.,h_i ~ CN(0,1)) and the channel noise between each secondary user and the target primary user are AWGN with zero mean and varianceσ_v² =1. For simplicity, we assume the channel gain {g } between secondary users and fusion center are constant _i AWGN (g_i =1), and the channel noise {n } between secondary users and fusion _i center are AWGN with zero mean and varianceσ_n² =1. The transmitted primary

signal has unit power s(k)² =1 and the detection interval is 2n samples. The proposed cooperation schemes are compared with selection combining method (SC i.e., selecting the user with maximum SNR), equal gain combination method (EGC

i.e., i M

w_i = 1M , =1,2,..., ) and single cognitive radio.

Secondly, we further consider the channel gain {g } between secondary users and _i fusion center are generated by a normal distribution with zero mean and unit variance, and they are assumed to be fixed during the detection interval. We assume that the channel noise between each secondary user to the target primary user and secondary users to fusion center are, respectively, AWGN with zero mean and varianceσ_v² =2

and 2σ_n² = . Under the condition, we observe the effect of different number of secondary user ( M ) and different variance of channel noise between each secondary user to the target primary user or secondary users to fusion center.

Fig. 3-2 The probability distribution function of the test statistics (u) under different hypotheses, with constant AWGN channel (g_i =1) ,M=3, n=50,σ_v² =1, and σ_n² =1. The result is the average of 100 simulations.

Fig. 3-3 The probability distribution function of the test statistics (u) under different hypotheses, with constant AWGN channel (g_i =1) ,M=10, n=50,σ_v² =1, and σ_n² =1. The result is the average of 100 simulations.

From figure 3-2 and figure 3-3, we show that the probability distribution functions of the test statistics under different hypotheses. We compare the distribution of optimization of modified reflection coefficient with the distribution of single cognitive radio SC. We can observe that the distance between uopt.PDF,H₀ and

, 1

.PDF H

uopt is larger than the distance between usc,H₀ and usc,H₁. Also, we can find that the spread of

, 1

.PDFH

uopt is narrower than that of

,H1

usc . On the other word, the variance of

, 1

.PDFH

uopt is smaller than the variance of

,H1

usc . Further, when the numbers of secondary user are increased, we can observe that the distance between

, 0

.PDFH

uopt and uopt.PDF,H1 become large.

According to above-mentioned, we obviously understand that the distribution of optimization of modified reflection coefficient and increased the numbers of secondary user would result in more accurate inference. These observations imply that the PDF optimization cooperation scheme outperforms any local spectrum sensing by individual secondary users.

Fig. 3-4 The probability of miss-detection (1−P_d) vs. the probability of false alarm (P_f ), with constant AWGN channel (g_i =1) ,M=3, n=50,σ_v² =1, and σ_n² =1. The result is the average of 1000 simulations.

Fig. 3-5 The probability of miss-detection (1−P_d) vs. the probability of false alarm (P_f ), with constant AWGN channel (g_i =1) ,M=10, n=50,σ_v² =1, and σ_n² =1. The result is the average of 1000 simulations.

From figure 3-4 and figure 3-5, we plot the probability of miss-detection (1−P_d) versus the probability of false alarm ( P ) under various approaches, such as _f optimized modified reflection coefficient method, equal gain combination method (EGC, the corresponding weight coefficient is expressed as i M

w_i 1M , 1,2,...,

=

= ),

selection combining method (SC, selecting the user with maximum SNR), and single cognitive radio. The probability of miss-detection (1−P_d) versus the probability of false alarm (P ) directly measures the interference level to the primary users for a _f

given P . The simulation shows that the proposed optimized modified reflection _f coefficient method (denoted as opt PDF) lead to much less interference (much higher probability of detection) to the primary user than single cognitive radio, selection combining method, and equal gain combination method. Also, we can find that the cooperation schemes (opt PDF, EGC) outperform single cognitive radio schemes (SC, single CR). The cooperation gain is due to control the combining weight coefficient which sharp the probability distribution function. Further, we can observe that when the numbers of secondary user are increased, the cooperation gain become large.

Fig. 3-6 The probability of miss-detection (1−P_d) vs. the probability of false alarm (P_f) under various M, with fading channel (g_iare generated according to a normal distribution and assumed to be fixed during a detection interval) , n=50,σ_v² =2, and σ_n² =2. The result is the average of 1000 simulations.

Fig. 3-7 The probability of miss-detection (1−P_d) vs. the probability of false alarm (P_f) under various (σ_v²,σ_n²), with fading channel (g_iare generated according to a normal distribution ) ,M=5, n=50. The result is the average of 1000 simulations.

From figure 3-6, we plot the probability of miss-detection (1−P_d) versus the probability of false alarm (P ) under different numbers of secondary user. We further _f consider the channel gain {g } between secondary users and fusion center are _i generated by a normal distribution with zero mean and unit variance, and they are assumed to be fixed during the detection interval. And, we observe that when the numbers of secondary user are increased, the performance become batter. In other words, under the same condition, the sensing reliability improves as the number of secondary users increase.

From figure 3-7, we plot the probability of miss-detection (1−P_d ) versus the probability of false alarm (P ) under different noise condition. And we consider the _f channel gain {g } between secondary users and fusion center are generated by a _i normal distribution with zero mean and unit variance, and they are assumed to be fixed during the detection interval. As we can observe, the detection performance degrades as the noise conditions become bad. We can also find that the channel noise

From figure 3-7, we plot the probability of miss-detection (1−P_d ) versus the probability of false alarm (P ) under different noise condition. And we consider the _f channel gain {g } between secondary users and fusion center are generated by a _i normal distribution with zero mean and unit variance, and they are assumed to be fixed during the detection interval. As we can observe, the detection performance degrades as the noise conditions become bad. We can also find that the channel noise

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