Optimal packet length for cognitive radio networks

Accepted Manuscript Optimal packet length for cognitive radio networks Ghassan Alnwaimi, Hatem Boujemaa

PII: DOI: Reference:

S1874-4907(18)30061-2 https://doi.org/10.1016/j.phycom.2018.10.004 PHYCOM 606

To appear in:

Physical Communication

Received date : 24 January 2018 Revised date : 3 October 2018 Accepted date : 19 October 2018 Please cite this article as: G. Alnwaimi, H. Boujemaa, Optimal packet length for cognitive radio networks, Physical Communication (2018), https://doi.org/10.1016/j.phycom.2018.10.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Optimal Packet Length for Cognitive Radio Networks Ghassan Alnwaimi *,Hatem Boujemaa ** (*) King Abdulaziz University, Kingdom of Saudi Arabia (**) University of Carthage, Sup’Com, COSIM Laboratory, Tunisia [email protected],[email protected], October 3, 2018

Abstract In this paper, we propose to enhance the throughput of Cognitive Radio Networks (CRN) by optimizing packet length. The throughput in the secondary and primary networks are maximized by optimizing packet length. The derived packet length takes into consideration the interference from primary transmitter to secondary receiver. We also take into account the interference from secondary source to primary receiver. For each average Signal to Noise Ratio (SNR), we provide the expression of packet length that yields to maximum throughput. The proposed optimal packet length for CRN allows 1-5 dB gains in terms of throughput with respect to fixed packet length.

Index Terms : Cognitive Radio Networks, Optimal packet length, Rayleigh fading channels. 1

1

Introduction

Cognitive Radio has gained much attention and prominence in the view of radio spectrum shortage. Adding cognitive features to the conventional networks allows more adaptation to the dynamic radio environment and introduces more spectral efficiency [1-3]. The overlay approach in cognitive radio networks allows cognitive users (i.e., secondary users SUs) to transmit at the same frequency spectrum as primary users (PUs), provided that secondary user dedicate part of its power to assist (relaying) primary transmission. The underlay approach allows SU to transmit over the same band as PU but the generated interference should be lower than a predefined threshold. In the interweave approach, SUs transmit only over frequency holes detected by spectrum sensing algorithms [4]. In traditional relay communications, cognitive relays can either relay the primary received signal using an amplify-and-forward (AF) or decode-and-forward (DF) protocol [5-13]. However, the deployment of such coexistence scenario between PUs and overlay/underlay SUs introduces more complexity. In spectrum sharing models, there is always a trade off between cautious protection margins for PUs and maximizing the efficiency of underutilized spectrum. The former protects license holders from unacceptable interference levels and guarantee the quality of service (QoS), while the latter is a requirement for the SUs, in order to have an efficient and reliable communication connections. Recent studies considered robust spectrum sensing algorithms in the presence of correlated antennas [14]. Optimal power allocation has been proposed in [15] to improve the detection performance of sensing algorithms. Spectrum allocation strategies using internet of things have been suggested in [16]. In previous studies, packet length is fixed in CRN

2

[1-3]. The scope of this paper is to optimize packet length for CRN to enhance the throughput in both primary and secondary networks. The derived optimal packet length takes into consideration both interferences from primary to secondary nodes as well as interference from secondary to primary nodes. Optimal packet length in non Cognitive Radio Networks has been derived in [17]. The analysis in [17] is valid for communications without interference between users. In this paper, we derive the optimal packet length in CRN taking into account the interference aspect. The derived optimal packet length for CRN allows 1-5 dB gains with respect to fixed packet length as considered in many papers [1-3]. For low average SNRs, packet length is reduced. At high average SNRs, packet length is increased to maximize the throughput. The proposed optimal packet length requires only a perfect knowledge of the average of absolute square of interference as well as the average SNR. For each average SNR, we derive the expression of the optimal packet length for both primary and secondary networks by taking into account the interfering signals. We have recently suggested in [18] an adaptive link layer protocol where packet length as well as the Modulation and Coding Schemes (MCS) are adapted to instantaneous SNR. The results of [18] are valid for Free Space Optical (FSO) communications when there are no interference. The paper is organized as follows. Next section provides the system model. Section 3 and 4 deal respectively with packet length optimization for secondary and primary networks. Section 5 gives some theoretical and simulation results. Conclusions are presented in section 6. 3

2

System Model

In the studied network shown in Fig. 1, there are a primary transmitter PT , a primary receiver PR , a secondary source S and a secondary destination D. Primary transmitter PT is always transmitting with a fixed power to Primary receiver PR . The secondary source transmits with a fixed power over the same channel as PR only when the generated interference to primary receiver PR is lower than interference threshold T : ISPR = ES |hSPR |2 is the generated interference from S to PR and should be lower than T . ES is the transmitted energy per symbol of S which is assumed to be constant, hSPR is the channel coefficient between S and PR . S transmits with a fixed power only when the generated interference to PR is lower than T . Otherwise, S remains idle. The secondary destination D receives interference from primary transmitter PT . Also, the primary receiver PR receives interference from secondary source S. Rayleigh fading channels between all nodes is assumed. The Probability Density Function (PDF) of absolute value of channel coefficient, rXY = |hXY |, between two nodes X and Y is given by [21] 2

frXY

2r − 2r (r) = 2 e σXY , ∀ r ≥ 0, σXY

2 2 where σXY = E(rXY ) and E(r) is the expectation of r.

4

(1)

PT

PR

S

D

Useful signal Interference

Figure 1: System model.

3

Packet Length Optimization in the Secondary Network

3.1

SINR CDF at Secondary Destination

It is assumed that secondary source has a fixed transmit power. When there is interference from primary node PT , the Signal to Interference plus Noise Ratio (SINR) between the secondary source S and secondary destination D is written as

ΓS,D =

ES |hSD |2 , EPT |hPT D |2 + N0

(2)

where ES is the transmitted energy per symbol of S which is assumed to be constant, N0 is the noise Power Spectral Density (PSD), hSD is the channel coefficient between 5

S and D, EPT is the transmitted energy per symbol of primary transmitter PT , hPT D is the channel coefficient between primary transmitter and node D. EPT |hPT D |2 represents interference at node D from PT . The Cumulative Distribution Function (CDF) of the SINR is written as [22] FΓSD (γ) = 1 −

N0 γ 2 − ES σSD ES σ 2 SD e 2 ES σSD + γEPT σP2 T D

=1−

γ 1 e− Γ , 1 + γα

(3)

where

2 σSD = E(|hSD |2 ),

(4)

σP2 T D = E(|hPT D |2 ),

(5)

E(X) is the expectation of X.

α=

EPT σP2 T D , 2 ES σSD

(6)

2 ES σSD . N0

(7)

and Γ=

3.2

Average Secondary Throughput

Each transmitted packet by the secondary source S contains N data bits and nd parity bits for error detection. These N + nd bits are converted in

N +nd log2 (M )

M-QAM symbols.

Using the results of [19], the Packet Error Probability (PEP) at secondary destination is upper bounded by P EPs < FΓSD (w0 ), 6

(8)

where index s refers to secondary PEP, w0 is a waterfall threshold given by [17] w0 = k1 ln(

N + nd ) + k2 , log2 (M )

(9)

where [17] 1 k1 = , c

k2 =

(10)

γe + ln(a) , c

(11)

γe ≃ 0.577 is the Euler constant, a and c are defined from the expression of Bit Error Probability (BEP) for M-QAM modulations [21]

BEP (γ) ≃

2(1 −

√1 ) M

log2 (M )

erf c

(√

3γ log2 (M ) 2(M − 1)

)

,

(12)

erf c(x) is the complementary error function,

2

erf c(x) ≤ e−x .

(13)

Using (12) and (13), the BEP is approximated by

BEP (γ) = ae−cγ ,

(14)

where,

a=

c=

2(1 −

√1 ) M

,

(15)

3 log2 (M ) . 2(M − 1)

(16)

log2 (M )

7

For M-QAM modulation, the number of received bits is N log2 (M ) and these bits are received with probability 1 − P EPs . The required transmit these bits is (N + nd )Ts and used bandwidth is B = 1/Ts where Ts is the symbol period. The secondary throughput in bit/s/Hz is written as

T hrs =

N log2 (M ) (1 − P EPs )P (ISPR < T ), (N + nd )Ts B

(17)

where B = 1/Ts is the used bandwidth and Ts is the symbol period. The term P (ISPR < T ) is due to the fact that secondary source is allowed to transmit only when it generate interference to primary receiver less than T. Finally, the secondary throughput is expressed as T hrs =

N log2 (M ) (1 − P EPs )P (ISPR < T ). N + nd

(18)

In fact, the secondary source transmits only when the generated interference to PR is lower than a predefined threshold T : P (ISPR < T ) = 1 − e

−

T ISP R

,

(19)

ISPR = ES |hSPR |2 is the generated interference from S to PR and ISPR = E(ISPR ). Using (2) and (7), the throughput is lower bounded by T hrs >

w0 − T N log2 (M ) 1 [1 − e ISPR ] e− Γ . N + nd 1 + w0 α

(20)

This lower bound is very tight [5]. Using (8), we can write −

T

k k N log2 (M ) [1 − e ISPR ] − 2 N + nd − 1 Γ ( Γ . T hrs > ) e N +nd N + nd 1 + k2 α + k1 α ln( log log2 (M ) ) (M ) 2

8

(21)

3.3

Optimal Packet Length for Secondary Cognitive Radio Networks : OPL-CRN

In the absence of primary interference, i.e., α = 0, the throughput is maximum for packet length equal to [17]

Nopt =

Γ × nd , k1

(22)

Γ is the average SNR and nd is the number of parity bits and k1 is defined in (9). Equation (21) provides the expression of Optimal Packet Length for Non CRN (OPLNCRN) as suggested in [17]. By setting to zero the derivative of (21), appendix A shows that we have the following equation f (N ) = a1 + a2 N + a3 ln(N + nd ) + a4 N ln(N + nd ) = 0,

(23)

a1 = nd + k2 αnd − αk1 nd ln(log2 (M )) − αnd k1 ,

(24)

where

a2 =

αk12 ln(log2 (M )) k1 αk1 k2 − αk1 − − , Γ Γ Γ

(25)

a3 = αk1 nd ,

(26)

αk12 . Γ

(27)

a4 = −

9

3.3.1

Newton Search Algorithm

Equation (23) can be solved iteratively using the Newton search algorithm that is initialized using the solution when there is no primary interference (22): N1 =

Γ × nd . k1

(28)

Only three iterations are performed to find the optimal packet length with good precision Ni+1 = Ni −

f (Ni ) , i = 1, 2, 3, f ′ (Ni )

(29)

where function f (x) has been defined in (23). Packet length is the closest integer to the solution find by iterations (23) so that N + nd is a multiple of the number of bits per M-QAM symbols log2 (M ).

3.3.2

Approximate Solution

At medium and high SNR, we have

a4 = −

αk12 << 1. Γ

(30)

If we neglect a4 N ln(N + nd ) in (23), we can write that

f (N ) = a1 + a2 N + a3 ln(N + nd ) ≃ 0.

(31)

Appendix B shows that optimal packet length is given by

Nopt =

a2 nd a2 − a1 a3 W (−1, e a3 e a3 ) − np , a2 a3

where W (−1, x) is the Lambert W function, an inverse of h(x) = xex . 10

(32)

For

−1 e

< y < 0, y = h(x) has two solutions. The larger solution is y = W (0, x)

and the smaller solution is W (−1, x). We have used the Matlab function W (−1, x) = Lambertw(−1, x) in order to obtain positive packet length in (32). We have used the nearest integer to (32) so that N + nd is a multiple of log2 (M ) (number of bits per M-QAM symbol). Equation (31) provides the expression of the proposed Optimal Packet Length for CRN (OPL-CRN). It requires the values of α and average secondary SNR Γ. An estimation of average SNR can be performed as described in [20]. The estimation of α can be obtained similarly. In fact, when the secondary source is idle (generate interference to PR is larger than T ), the received signal at secondary destination can be written as yD =

√

EPT xhPT D + n where x is the transmitted symbol by PT and n is additive white

gaussian noise. yD can be used to estimate the average of absolute square of primary interference EPT E(|hPT D |2 ) similarly to [20] which is needed to compute α.

4 4.1

Optimal Packet Length for Primary Network SINR CDF at Primary Receiver

It is assumed that the power of secondary source is fixed. S transmits only when the generated interference is lower than T . The CDF of SINR at primary receiver can be expressed as ( ) P (ΓPT PR < x) = P ΓPT PR < x|ES |hSPR |2 < T ( ) ×P ES |hSPR |2 < T

( ) ( ) +P ΓPT PR < x|ES |hSPR |2 > T P ES |hSPR |2 > T .

11

(33)

When the interference is larger than T , ES |hSPR |2 > T , the secondary source S is idle and there is no interference so that we can write

( ) −N0 x P ΓPT PR < x|ES |hSPR |2 > T = 1 − exp( ). EPT σP2 T PR

(34)

When the interference is lower than T , ES |hSPR |2 < T , the secondary source S transmits and we can write

( ) ( ) P ΓPT PR < x|ES |hSPR |2 < T P ES |hSPR |2 < T =

[ 1 − exp(−

] [ ] EPT σP2 T PR T N0 x ) × 1 − exp(− ) . 2 2 ES σSP EPT σP2 T PR EPT σP2 T PR + xES σSP R R

(35)

The proof is provided in appendix C. If we replace T by +∞, we obtain the same expression as (2). Using (33), (33) and (34), the Cumulative Distribution Function of SNR is given by FΓPT PR (x) = P (ΓPT PR

] T −N0 x ) exp(− ) < x) = 1 − exp( 2 2 EPT σPT PR ES σSP R [

[

] [ ] EPT σP2 T PR T N0 x + 1 − exp(− ) × 1 − exp(− ) . 2 2 ES σSP EPT σP2 T PR EPT σP2 T PR + xES σSP R R

4.2

(36)

Average Primary Throughput

Each transmitted packet by the secondary source S contains N data bits and nd parity bits for error detection. These N + nd bits are converted in

12

N +nd log2 (M )

M-QAM symbols.

Using the results of [19], the PEP at secondary destination is upper bounded by P EPp < FΓPT PR (w0 ),

(37)

where index p refers to primary network, w0 is a waterfall threshold defined in (8). Using the expression of w0 (8), the PEP is expressed as P EPp < exp(− +[1 − exp(−

k2 N + nd − Γk1p T )[1 − exp(− )( ) ] 2 ES σSP log (M ) Γ 2 p R

T k2 N + nd − Γk1p 1 )][1 − exp(− )( ) ], 2 N +nd ES σSPR 1 + k2 β + k1 β ln( log ) Γp log2 (M ) (M )

(38)

2

where Γp is the average SNR at the primary receiver PR defined as Γp = β=

EPT σP2 T PR , N0

(39)

2 ES σSP R . EPT σP2 T PR

(40)

The primary throughput for M-QAM modulation is equal to T hrp =

N log2 (M ) (1 − P EPp ). N + nd

(41)

Using (37), we can write T hrp >

N log2 (M ) k2 N + nd − Γk1p exp(− )( ) N + nd Γp log2 (M )

1 − exp(− ES σT2 ) T SPR ×[exp(− )+ ]. 2 N +nd ES σSPR 1 + k2 β + k1 β ln( log ) (M )

(42)

2

The optimal packet length maximizes the throughput. A closed form expression can not be obtained. We can use the Gradient search algorithms that consists to compute the optimal value of packet length N iteratively.

Ni+1 = Ni + µ

∂T hrp (N = Ni ) . ∂N 13

(43)

Usually, the gradient search is used to minimize a function f and we write, Ni+1 = Ni −µ ∂f (N∂N=Ni ) . Here, we aim to maximize the throughput which is equivalent to minimize f = −T hrp . The expression of

∂T hrp ∂N

is provided in appendix D. During simulations, we used the

following values, µ = 1, and initialization N0 is given in (21) which is the optimal packet length for Non CRN. Then, some iterations are performed until convergence |Ni+1 −Ni | < ε with ε = 0.1. Packet length is the closed integer to the iterative solution obtained by gradient search so that N + nd is a multiple of log2 (M ), the number of bits per M-QAM symbol.

5

Theoretical and Simulation Results

Fig. 2 and Fig. 3 show the throughput in secondary network for interference threshold T = 1, ISPR = E(ISPR ) = 1 and a QPSK modulation M = 4. Simulations were performed using Matlab Software. The transmitted energy per symbol ES = EPT = 1 and σSD = 1. The value of σP2 T D = 0.25 ( α = 0.25 low primary interference) and σP2 T D = 0.5 (α = 0.5 high primary interference). We first notice that the proposed packet length OPL-CRN (31) allows similar results to Newton search described in section 3.3.1. Also, the optimal packet length for Non CRN OPL-NCRN derived in [17] and given in (21) cannot be applied for CRN. In fact, at low SNRs, we can neglect the primary interference so that (31) and (21) offer the same performance. However, at medium and high SNRs, the primary interference cannot be neglected and (21) offers bad

14

performance. The proposed optimal packet length offers 3-5 dB gains with respect to fixed packet length N = 290 for nd = 10. 1.4

1.2

α=0.25 Low primary interference

Secondary Throughput

1

0.8

0.6

0.4

Adaptive Packet Length : Theory Fixed Packet Length : N=290, Theory Adaptive packet length : OPL−CRN (26), Sim Fixed packet length : N=290, Sim Adaptive Packet Length : Newton Search, Sim Adaptive packet length : OPL−NCRN (16), Sim

0.2

0

0

2

4

6

8

10 Eb/N0 (dB)

12

14

16

18

20

Figure 2: Secondary Throughput using fixed and adaptive packet length for QPSK modulation : Low primary interference.

15

0.9

0.8

0.7

Secondary Throughput

0.6

0.5

0.4 Adaptive Packet Length : Theory Fixed Packet Length : N=290, Theory Adaptive packet length : OPL−CRN (26), Sim Fixed packet length : N=290, Sim Adaptive Packet Length : Newton Search, Sim Adaptive packet length : OPL−NCRN (16), Sim

0.3

0.2

α=0.5 High primary interference

0.1

0

0

2

4

6

8

10 E /N (dB) b

12

14

16

18

20

0

Figure 3: Secondary Throughput using fixed and adaptive packet length for QPSK modulation : High primary interference.

Fig. 4 shows the secondary throughput versus packet length. We notice that simulation results are congruent with the theoretical derivations. By optimizing packet length, the throughput is significatively increased.

16

1 Theory Sim 0.9

0.8

Secondary Throughput

0.7

0.6

0.5

0.4 α=0.25 Low primary interference, Eb/N0=10dB 0.3

0.2

0.1

0

50

100

150

200 250 300 Packet Length N

350

400

450

500

Figure 4: Secondary Throughput versus packet length N : Eb /N0 = 10dB.

Fig. 5 shows the primary throughput versus packet length for β = 0.25 and interference threshold T = 1. We notice that simulation results are congruent with the theoretical derivations. By optimizing packet length, the primary throughput is significatively increased.

17

0.35 Theory Simulations 0.3

Primary throughput

0.25

0.2

0.15

0.1

0.05

0

0

100

200

300

400 500 600 Packet Length N

700

800

900

1000

Figure 5: Primary Throughput versus packet length N : Eb /N0 = 5dB.

Fig. 6 shows the throughput in primary network for interference threshold T = 1 and a QPSK modulation M = 4. The transmitted energy per symbol ES = EPT = 1, σP2 T PR = 1 2 and σSP = 0.5. Therefore, β = 0.5. The proposed optimal packet length offers 1.1 dB R

gains with respect to fixed packet length N = 290 for nd = 10. Optimal packet length has been obtained using the Gradient search algorithm and offers better performance than N =290, 490 and 90 as considered in [23-26]. N=90 offers larger throughput than N = 290 at low SNR. However, N = 90 is not a good choice at high SNR.

18

1

0.9

0.8

Primary throughput

0.7

0.6

0.5

0.4

0.3 Simulations : Optimal packet Length Simulations : Fixed Packet Length N=90 Simulations : Fixed Packet Length N=290 Simulations : Fixed Packet Length N=490

0.2

0.1

0

0

2

4

6

8

10 Eb/N0(dB)

12

14

16

18

20

Figure 6: Primary Throughput using fixed and adaptive packet length for QPSK modulation.

Fig. 7 shows the performance of optimal packet length with known and estimated average SNR and β. We used 4 pilot symbols per packet and estimates where averaged over 10 consecutive packets. When the SNR and β are estimated, we obtain close performance ideal case with known parameters. The estimation of these parameters is performed as follows. When the seconder source is idle, the received signal at PR is given by

yPR (i) =

√ EPT xi hPT PR + ni ,

(44)

where i is symbol index, xi is the transmitted symbol by PT and ni is additive white gaussian noise. 19

We use P pilot symbols (known training symbols) placed at the beginning of each packet to compute P √ 1 ∑ yPR (i)x∗i ≃ EPT hPT PR , V = P i=1

(45)

where x∗i is complex conjugate of xi and |xi | = 1. Then we make an average of |V |2 over 10 consecutive packets to estimate EPT σP2 T PR . To estimate the average SNR in primary network, Γp given in (38), we need also to estimate the noise PSD N0 :

P √ 1 ∑ ˆ P P xi |2 . ˆ N0 = |yPR (i) − EPT h T R P i=1

(46)

The symbol energy of pilot symbol is assumed to be known. The estimate of noise PSD, ˆ P P is an estimate of channel coefficient Nˆ0 , is averaged over 10 consecutive packets. h T R obtained from ˆP P = h T R

P ∑ 1 √ yPR (i)x∗i . P EPT i=1

(47)

The estimation of α and average SNR in secondary network can be performed similarly.

20

1

0.9

0.8

Primary throughput

0.7

0.6

0.5

0.4 Estimated SNR and β Perfect SNR, β

0.3

0.2

0.1

0

0

2

4

6

8

10 Eb/N0(dB)

12

14

16

18

20

Figure 7: Primary Throughput using adaptive packet length for QPSK modulation : known and estimated average SNR and β.

6

Conclusions

In this paper, we optimized packet length in order to obtain the highest throughput in the secondary and primary networks of CRN. The analysis takes into consideration the interference from primary and secondary transmitter. For each average SNR, we provide the expression of optimal packet length. Optimal packet length allows 1-5 dB gains in terms of throughput with respect to fixed packet length. We have also shown that optimal packet length derived in [17] cannot be used in secondary network because it does not take into consideration the interfering signal from primary user. Also, the results of [17] 21

cannot be used for primary network of CRN because it does not take into consideration the secondary interference at primary receiver. Acknowledgments This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, under grant no. G 613 / 315 / 1439 . The authors, therefore, acknowledge with thanks the DSR for technical and financial support. Appendix A : The derivative of secondary throughput with respect to packet length is given by

−

T

k k ∂T hrs nd log2 (M ) [1 − e ISPR ] − 2 N + nd − 1 Γ ( Γ e ≃ ) N +nd ∂N (N + nd )2 1 + k2 α + k1 α ln( log log (M ) ) 2 (M ) 2

−

T

−

T

k k1 −k1 [1 − e ISPR ] k1 N log2 (M ) − 2 −1 Γ log (M ) Γ (N + n ) Γ e − d 2 N +nd Γ N + nd 1 + k2 α + k1 α ln( log (M ) ) 2

k k [1 − e ISPR ]α kN1 N log2 (M ) − 2 N + nd − 1 Γ ( Γ = 0. − e ) N +nd 2 N + nd [1 + k2 α + k1 α ln( log log (M ) )] 2 (M )

(48)

2

After some simplifications, we have nd N k1 αk1 − − = 0. N +nd N + nd (N + nd )Γ 1 + k2 α + k1 α ln( log ) (M )

(49)

2

The last equation can be written as (22). Appendix B : We have f (N ) = a1 + a2 N + a3 ln(N + nd ) ≃ 0.

(50)

Let z = N + nd , we deduce a1 nd a2 za2 − + + log(z) ≃ 0. a3 a3 a3 22

(51)

We deduce

za2 za2 nd a 2 a 1 a2 + ln( )≃ − + ln( ). a3 a3 a3 a3 a3

(52)

We take the exponential of the last equation to write h(

nd a2 a za2 − 1 a2 ) ≃ e a3 e a3 , a3 a3

(53)

where h(x) = xex . We deduce the optimal packet length

Nopt =

a3 a2 nd a2 − a1 W (−1, e a3 e a3 ) − np , a2 a3

(54)

where W (−1, x) is the Lambert W function, an inverse of h(x) = xex . Appendix C We have

P (ΓPT PR < x, |hSPR |2 <

T T ) = P (EPT |hPT PR |2 < x(N0 + ES |hSPR |2 ), |hSPR |2 < ). ES ES

Let Z = EPT |hPT PR |2 and W = N0 + ES |hSPR |2 | |hSPR |2 <

T P (EPT |hPT PR | < x(N0 + ES |hSPR | )| |hSPR | < )= ES 2

2

2

∫

T , ES

(55)

we deduce

N0 +T N0

FZ (xu)fW (u − N0 )du, (56)

where 0) exp(− E(u−N ) 2 S σSP R [ fW (u − N0 ) = 2 ES σSP 1 − exp(− ES σT2 R

SPR

23

], )

(57)

FZ (xu) = 1 − exp(−

xu ). EPT σP2 T PR

(58)

We finally obtain P (ΓPT PR

[ ] [ ] 2 E σ T T N x P T PT PR 0 < x, |hSPR |2 < ) = 1 − exp(− ) × 1 − exp(− ) . 2 2 2 2 ES ES σSP E σ E σ + xES σSP P P T PT PR T PT PR R R (59)

Appendix D Using (41), the derivative of primary throughput with respect to packet length can be written as 1 − exp(− ES σT2 ) ∂T hrp nd log2 (M ) k2 N + nd − Γk1p T SPR = exp(− )( ) ×[exp(− )+ ] 2 N +nd ∂N (N + nd )2 ES σSP Γp log2 (M ) 1 + k β + k β ln( ) 2 1 R log (M ) 2

1 − exp(− ES σT2 ) k1 N k2 N + nd − Γk1p −1 T SPR ] − exp(− )( ) × [exp(− )+ 2 N +nd log (M ) E σ ) Γp (N + nd ) Γp 1 + k2 β + k1 β ln( log S SPR 2 (M ) 2

1 − exp(− ES σT2 ) k1 βN log2 (M ) k2 N + nd − Γk1p SPR − exp(− )( ) ×[ ]. N +nd 2 (N + nd ) Γp log2 (M ) (1 + k2 β + k1 β ln( log ))2 (M )

(60)

2

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Biograaphy

mi (M’14) received the B.Sc. degree (firrst honor) in electron nics and Ghassaan Alnwaim communnication enngineering from Kingg Abdulazizz University y, Jeddah, Saudi Araabia, the M.Sc. ddegree (withh distinction n) in mobilee and satellitte communications froom the Univ versity of Surrey, Guildford, U.K., and the t Ph.D. ddegree from the Institutte for Comm munication Systems (formerrly Centre foor Commun nication Sysstems Reseaarch), Univeersity of Suurrey, in 201 14. He is an Assiistant Profeessor with the t Departm ment of Eleectrical and Computer Engineerin ng, King Abdulazziz Universsity. His ressearch interrests includ de the desig gn of cogniitive radio systems, self-orgganization networks, n spectrum s s ensing, speectrum man nagement, dynamic spectrum s access ttechniques, and interferrence managgement in femtocells f and a heterogeeneous netw works.

Hatem Boujemaaa (M'02) waas born in T Tunis, Tuniisia. He recceived the E Engineer’s Diploma D from '' Ecole Polyytechnique de d Tunis'', in 1997, th he MSC in digital com mmunicatio ons from 8 and the Phh. D. degreee in electron nics and com mmunicatio ons from ''Telecom Paris Tecch'', in 1998 me universityy in 2001. From F Octobber 1998 to o Septemberr 2001, he pprepared hiis Ph. D. the sam degree at France Telecom R&D, Issyy-les-Mouliineaux, Fraance. Durinng this perriod, he participated in thee RNRT pro oject AUBE E. From October O 200 01 to Januaary 2002, he h joined ''Ecole Supérieure d'Electricitté'', Gif-sur--Yvette, France, and worked w on mobile localization RT project LUTECE. In I Februaryy 2002, he joined j SUPCOM wherre he is a Prrofessor. for RNR His reseearch activiities are in the field oof digital communications, DS-C CDMA, OFD DM and MC-CD DMA systeems, HARQ Q protocolls, Cooperaative Comm municationss, Cognitiv ve radio networkks, Scheduliing, Synchrronization, N Network planning, Info formation Thheory, Equalization and Anttenna Proceessing.