On Outage and Interference in 802.22 Cognitive Radio Networks under Deterministic Network Geometry S. A. R. Zaidi1 , B. Zafar2 , D. C. McLernon1 , M. Ghogho1 1
University of Leeds, Leeds, United Kingdom Ajou University, Suwon, Republic of Korea {elsarz, d.c.mclernon, m.ghogho}@leeds.ac.uk,
[email protected] 2
Abstract In this paper, we consider a 802.22 type cognitive radio network. We assume that, the communication in such a network is subjected to the small scale fading, modeled by Rayleigh distribution. Considering the deterministic network geometry we develop the analytical framework to characterize the interference and outage of the primary network.
path loss model and contribution of secondary cells in 1st tier across primary network. To the best of our knowledge none of the prior studies have modeled interference in actual deterministic geometry based 802.22 CRN under fading conditions. In this paper, we establish the interference model for 802.22 CRN under Rayleigh fading environment. 1.1
1.
Introduction
Recently, there has been a considerable interest in developing intelligent radio networks. These intelligent or Cognitive Radio Networks(CRNs) are capable of utilizing the scarce wireless specturm in optimum manner. Regulatory bodies such as FCC and Ofcom have already reported that specturm scarcity is essentially the consequence of the poor utilization of frequency bands across space and time. CRNs are envisoned to exploit the vacant frequency bands across space and time in opportunistic manner. In this regards, IEEE has already formed the 802.22 workgroup to develop an air interface for Dynamic Specturm Access (DSA) in TV frequency band. CRNs, frequently refered as a secondary networks can opportunistically exploit specturm vacincies provided that they do not interfere with the primary or legacy network. Regulatory bodies such as FCC have introduced ‘Interference temprature’ as a decisive measure for the secondary network’s transmission. Several opportunistic schemes for utilizing specturm vacaincies are proposed in literature. Essentially all of them can be classified in to three classes: Overlay specturm access, Underlay specturm access and Interweave approach [3]. However, none of these schemes provide interference free transmission. This is indeed consequence of random channel impairment process (multipath fading and large scale shadowing). Hence, in order to accurately quantify the performance of different wireless protocols, the knowledge of the interference distribution is essential. In past several studies [2, 5] have reported interference distribution for cognitive ad hoc network. Unfortunately these studies can not be generalized for 802.22 type network model. Shankar et al. in [4] has presented the interference model considering the simple
Contributions
Our contribution is three fold: 1. We argue that only considering the contribution of the 1st tier of secondary network across primary network is not sufficient. We determine the number of tiers that should be considered and generalize [4] to determine the total number of interferer’s at any arbitrary primary receiver. 2. We then determine the distribution of Interference at any arbitrary primary receiver. We also propose how overall network outage can be established from individual receiver outage. 3. Lastly, we show that the interference distribution is skewed and hence considering Gaussian distribution is in appropriate.
Symbol PT X i PRX STj X rs rp ∆ re |hij |2 dj di dij Ps
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No NIN T
Definition primary transmitter ith primary receiver j th secondary receiver radius of secondary’s cell radius of the primary’s cell primary’s guard zone radius of the pirmary’s exclusion region re = rp + ∆ channel gain from j th secondary transmitter to ith primary receiver distance from j th secondary transmitter to PT X distance of ith primary receiver from primary transmitter distance of ith primary receiver from j th secondary transmitter transmission power of any secondary transmitter noise power number of interferers
2.
Network Model
This regulatory constraint then impose restriction of the form
Consider a primary network with the transmitter PT X centered at the origin. The ith primary receiver lies at the distance di from PT X inside the primary network of the radius rp . Several secondary cells of the radius rs surround the primary network. These secondary cells form several concentric annulus such that, nth annulus has an internal radius re + 2nrs and the external radius of re + 2(n + 1)rs , where n = 0, 1, ..., ∞ and re is the radius of exclusion region/guard zone for the primary network. For mathematical simplicity, it can be safely assumed that re is multiple of the secondary cell’s radius rs ( i.e., αrs ) ∆ + rp α= α≥1 (1) rs
Pr{Ii < Ith } ≥ β
(5)
th
Hence outage probability of the i primary receiver is given by (i)
Pout = 1 − Pr{Ii < Ith } ≤ 1 − β
(6)
Considering (6) we can define the primary network’s outage probability in several ways. Here we discuss two such ways 1. K out of M : The primary network is considered to incur outage if K out of M primary receivers incur outage, i.e., M PN Pout = (1 − β)K β M −K (7) K 2. Minimum Interference: The primary network is considered to experience outage if Pr{min(Ii ) > Ith } i
Or an alternative definition can be given as, (i)
PN Pout = max Pout
(8)
i
The main challenge now is the determination of NIN T , the probability density function(PDF) and the cumulative density function (CDF) of Ii . Note that the network model studied here, in practise models 802.22 WRAN, where a single DTV transmitter and multiple DTV receivers form primary network.
3.
Figure 1. Network Model The channel coefficient from the j th secondary transmitter to the ith primary receiver hij is asummed to be Rayleigh distributed with C (2) dkij where, C is the frequency dependent constant, dij is the distance from the j th secondary transmitter to the ith primary receiver and k is the environment dependent path-loss exponent, generally 2 ≤ k ≤ 4 . By using law of cosines q (3) dij = d2i + d2j − 2di dj cos θ E(|hij |2 ) =
Fig. 1 provides graphical illustration of the relationship between di , dj and dij . Moreover, channel coefficients are assumed to be identical and independently distributed (i.i.d) for different links. All secondary transmitters are assumed to transmit with the same power Ps 1 with opportunistic underlay specturm access mechanism. The aggregate interference power Ii due to the presence of NIN T secondary interferers at the ith primary receiver is then given by NIN T
Ii =
X P |hij |2 No j=1
(4)
Considering the ‘Interference temprature’ constraint as in [1, 2] the ith receiver can tolerate the maximum aggregate interference of Ith . 1A
more complex scheme can consider power allocation, in such a case P generally a constraint of the form i Pi ≤ P exists, thus our analytical framework can be easily adapated.
Determination of the Number of Interferers NIN T .
The determination of the effective number of interferers NIN T , is essentially the circle packing problem in annulus. However one should also note that the primary network is surrounded by the infinite number of concentric annulus, this makes the analysis difficult and mathematically intractable. Our main objective in this section is to extend the method of [4], and develop a generalized formula th n to determine the number of interferer NIN T present in the n annulus. 3.1
Number of Interferers in the nth annulus.
By geometry (See Fig. 2) the angle ψ n subtended by the secondary cell in the nth annulus at the primary network’s origin is given by rs ψ n = tan−1 (9) re + (2n + 1)rs since re = αrs ψ n = tan−1
1 α + 2n + 1
(10)
n Hence NIN T is given by n NIN T =
tan−1
π
1 α+2n+1
(11)
As stated in 802.22 WRAN standarad and as depicted by the calculations under simple path loss model in [4] re = 150 km while rs = 33 km generally, i.e. re ≈ 5rs . Realizing that two DTV networks can peacefully coexist if their transmitters are 2re times apart. The number of tiers or annulus which can contribute to the interference can be estimated as re /rs 5 m= = =3 (12) 2 2 Consequently, considering only inteference contribution of the1st annulus as in [4] is not sufficient. The total number of interferers is
and di is randomly selected inside primary network . Fig 3 depicts the interference histogram.
Figure 2. Relationship between ψ n , rs and re given by NIN T =
m−1 X
n NIN T
(13)
n=0
Figure 3. Interference Histogram for 10,000 iterations
4.
Interference Distribution
Since we determined the total number of interferers in last section, we now focus our attention towards establishing the closed form expression for PDF of aggregate inteference at the ith primary receiver. As discussed earlier, we assumed that the small scale fading is accurately characterized by the Rayleigh distribution, Consequently we know that channel gain |hij |2 is exponentially distributed with mean (1/λij ) λij =
No Ps E(|hij |2 )
(14)
By our knowledge of probability, we know that sum of exponential random variables with different means possesses the hypoexponential distribution or phase type distribution. Hence the distribution of the interference received at the ith primary receiver (Ii ) is given by NIN T
fIi (x) =
X
Cj,NIN T λij exp (−λij x)
(15)
j=1
where Cj,NIN T =
Y k6=j
λik λik − λij
The outage probability is then given as NIN T (i)
Pout = 1 −
X
Cj,NIN T exp (−λij Ith )
(16)
j=1
As explained earlier the outage probability of the primary network can be easily computed from (16).
5.
Discussion
The analytical machinery developed in the last section forms a comprehensive framework for the performance analysis of 802.22 CRNs. Nevertheless simulation results are essential to make necessary inferences. Due to the lack of space simulation results for the CDF and PDF (Hypoexponential distribution) of interference are not provided here. However, in this section we highlight some interesting results. Note that the network model presented in Fig. 1 is actually generated by MATLAB simulation of expression derived in previous section. This verifies the validity of our analytical expressions. Utilizing the developed network model, we perform MonteCarlo simulation for interference recieved under γ = PNsoC = 20dB
From Fig. 3 we observe that the interference histogram is skewed. Hence interestingly the interference distribution even under deterministic geometry is skewed, i.e., assuming the Gaussian distribution for the interference power as in [5] is too optimistic. In case of random networks, it is very well established that the interference distribution at transmitter centred at the origin is α stable(heavy tailed) . Often it is considered that the fact that number of transmitters in such a network is infinite accounts for skeweness and stable distribution. However as shown here, it is not necessarily the number of transmitter that can lead to skewness and stable distribution, indeed uncertainites such as small scale fading contribute signficantly towards heavy tailed behavior.
6.
Conclusion
In this paper, we presented a novel mathematical framework to determine the interference distribution in 802.22 CRNs under deterministic network geometry. Our proposed analytical machinery inherently caters for the multipath impairment process by considering Rayleigh fading environment. We also showed that considering the interference contribution from only immediate neighbours (i.e. secondary network in 1st annulus) is not sufficient and at least 3 annulus should be considered for interference modeling . Moreover, we presented the novel closed form expressions for the PDF and CDF of the interference distribution. We also discussed several possible ways to characterize network outage probability from individual receiver outage probabilites. Lastly, we demonstrated that even under deterministic network geometry the interference distribution is skewed, i.e., it is too optimistic to assume guassian distribution.
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