Square Law Selection Diversity for Wideband Spectrum Sensing Under Fading Kamal M. Captain
Manjunath V. Joshi
Dhirubhai Ambani Institute of Information and Communication Technology Gujarat, India 382007 Email:
[email protected]
Dhirubhai Ambani Institute of Information and Communication Technology Gujarat, India 382007 Email: mv
[email protected]
Abstract—In this paper we study the use of diversity for performance improvement in wideband spectrum sensing under fading. A new detection algorithm, namely, ranked square law selection (R-SLS) is proposed. The performance of the proposed algorithm is carried out under Nakagami fading. The analysis shows that the proposed algorithm outperforms ranked channel detection (RCD) algorithm without diversity. Analytical results are verified by Monte Carlo simulations. Effect of different parameters on the performance of proposed algorithm is also studied.
I. I NTRODUCTION Cognitive radio (CR) can provide solution to the problem of underutilization of licensed bands by allowing the secondary users (SU) to access the licensed bands when the primary users (PU) are inactive. This is done by finding occupancy status of primary users which is termed as spectrum sensing (SS) and is one of the main tasks for CR [1]. Since the licensed bands are underutilized, spectrum occupancy of primary users is sparse. If the secondary users can sense wide frequency bands at a time, they can provide multiple spectrum opportunities. The Nyquist sampling frequency required to sense wideband would be very high and practically challenging. Significant amount of work has been done on the wideband spectrum sensing (WSS) using the concept of compressed sensing (CS) [2]– [5] to reduce the sampling rate for WSS. However CS based sampling schemes are complex and are expensive. In general SUs may not be interested in finding all the spectrum opportunities, instead they may be interested in finding sufficient number of spectrum opportunities. In [6], partial band Nyquist sampling (PBNS) scheme is proposed in which instead of the entire wideband, only a part of it is sampled thus reducing the sampling rate. PBNS uses the traditional Nyquist sampling and hence represents the simplest WSS scheme. Authors in [6], have proposed two detection algorithms for WSS. These include channel by channel detection (CCD) and ranked channel detection (RCD) in which diversity is not used. They modeled primary signal as well as noise as additive white Gaussian (AWG). Very limited work has been done on WSS considering the fading [7], [8]. In [9], performance of these detection algorithms for two different fading channels, namely, Rayleigh and Nakagami fadings are analyzed. Although many researchers have used diversity
reception to combat effect of fading and shadowing, its use has been restricted to narrowband spectrum sensing [10]– [14]. It is interesting to see that performance of WSS under fading can be improved by using diversity reception. Here in this work, we propose a new detection algorithm, namely, ranked square law selection (R-SLS), which uses diversity to improve the performance of WSS under fading. We use PBNS scheme and metrics proposed in [6] to quantify the performance of the proposed algorithm. Due to the page limitations, we provide mathematical analysis of the proposed algorithm by considering Nakagami fading only. Nakagami fading is a generalized fading model and includes Rayleigh, Rice and Hoyt fading models as its special cases. Mathematical analysis is verified by Monte-Carlo simulation. We also study the effect of different parameters on the performance of the proposed algorithm using simulations. We observe that proposed algorithm outperforms RCD without diversity. II. P RIMARY S YSTEM M ODEL AND P ERFORMANCE M ETRICS The wideband channel is modeled as the collection of 𝑀 consecutive narrowband channels each of bandwidth 𝐵0 and hence it has a total bandwidth of 𝐵 = 𝑀 𝐵0 , [5], [6], [15]– [18]. The primary transmission within each band is subjected to fading. We assume that the probability of occupancy of each narrowband channel is 𝑝 and the occupancy status remains unchanged during the observation interval. Although the channels in wideband may be correlated in practice, we assume that they are independent. In order to detect the free channels the detector at the SU has to decide between presence and absence of the primary users with the hypotheses 𝐻𝑚 = 1 and 𝐻𝑚 = 0, respectively, for the 𝑚𝑡ℎ channel. The received signal at the SU for each narrowband channel is modeled as 𝑦(𝑡) = ℎ ⋅ 𝑠(𝑡) + 𝑛(𝑡), (1) where 𝑠(𝑡) is the primary signal, 𝑦(𝑡) is the received signal at the SU, 𝑛(𝑡) is the additive white Gaussian noise (AWGN) with 0 mean and variance 1 and ℎ is the channel gain. In this work we used two recently proposed performance metrics: probability of insufficient spectrum opportunity (𝑃𝐼𝑆𝑂 ) and probability of excessive interference opportunity (𝑃𝐸𝐼𝑂 ) [6]. They are defined as 𝑃𝐼𝑆𝑂 = 𝑃 𝑟 {𝑆 < 𝑆𝑑 }
978-1-5090-1701-0/16/$31.00 ©2016 IEEE
and 𝑃𝐸𝐼𝑂 = 𝑃 𝑟 {𝐼 > 𝐼𝑑 }, respectively. Here, 𝑆 and 𝑆𝑑 represent number of successfully identified and desired free (OFF) channels, respectively. Similarly, 𝐼 and 𝐼𝑑 correspond to the number of missed ON channels (ON channels declared OFF) and allowed number of interference to the primary users, respectively. 𝑃 𝑟 {⋅} represents the probability operator. III. R ANKED S QUARE L AW S ELECTION (R-SLS) D ETECTION Fig. 1. represents the block diagram of the detection scheme. There are 𝑃 number of receiver antennas (diversity branches) and 𝑀 narrowband channels. The PBNS filters out 𝐿 channels out of the total of 𝑀 channels and samples them at the corresponding Nyquist rate of 2𝐿𝐵0 . Considering that 𝑁 number of samples per channel are taken, the number of samples in the sensing window will be 𝑁 𝐿. Let 𝑉 [𝑘], 0 ≤ 𝑘 ≤ 𝑁 𝐿 − 1 denote the normalized discrete Fourier transform (DFT) of the received wideband signal of 𝐿 channels. Then, the energy for the 𝑚𝑡ℎ narrowband channel which is a random variable can be obtained as ∑ 2 ∣𝑉 [𝑘]∣ , (2) 𝑇𝑚 =
As shown in Fig. 1, energy computed within each of the 𝐿 channels from all the diversity branches are given as input to a selector. The selector selects highest energy for each of the 𝐿 channels. Once, this is done, the selected channels are ranked, i.e., arranged in ascending order with regard to 𝑇𝑚 , 𝑚 = 1, 2, ⋅ ⋅ ⋅ , 𝐿. Then the decision is taken on first 𝐿𝑑 channels, where 𝐿𝑑 ≤ 𝑆𝑑 ≤ 𝐿. Remaining 𝑀 −𝐿𝑑 channels are ignored. Fig. 1 represents the case with 𝐿𝑑 = 2. Here, 𝐶1 , 𝐶2 , ⋅ ⋅ ⋅ , 𝐶𝐿 represent 𝐿 narrowband channels. As shown in Fig.1, diversity 𝑃 1 and 𝑇𝑚𝑎𝑥 branches 𝑃 and 1 have maximum energies 𝑇𝑚𝑎𝑥 1 2 𝑗 for channels 𝐶1 and 𝐶2 , respectively. Note that, in 𝑇𝑚𝑎𝑥𝑖 , 𝑚𝑎𝑥𝑖 and 𝑗 represent the selected maximum energy in channel i and the corresponding diversity branch j, respectively, where 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝐿 and 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑃 . After ranking 𝐶𝐿 is shown to have the lowest energy and 𝐶2 has next lowest energy from the selected maximum energies. The detection is made on these channels 𝐿 and 2. Based on threshold 𝜏 , 𝐶𝐿 and 𝐶2 are shown as free and occupied, respectively. Note that reducing the number of diversity branches to one results in RCD proposed in [6]. The steps involved in detection using a fixed threshold 𝜏 are listed in Algorithm 1.
𝑘∈𝐼𝑚
where, 1 ≤ 𝑚 ≤ 𝐿 and 𝐽𝑚 is the set of frequency indices that fall into channel 𝑚.
IV. P ERFORMANCE A NALYSIS OF R-SLS U NDER NAKAGAMI FADING In this section we provide the mathematical analysis of the proposed R-SLS detection algorithm. Since, carrying out the analysis for general case is involved, we consider a simple case with 𝐿 = 𝑃 = 2, 𝐿𝑑 = 1 = 𝑆𝑑 = 1, 𝐼𝑑 = 0 and same average signal to noise ratio (SNR) at the input of all diversity branches. However, this can be extended to general case. The probability density function (pdf) of received energy under Nakagami fading is given as [19], { 𝑡 1(𝑡)𝑡𝐺−1 𝑒− 2 𝐹 (𝑚 ; 𝐺; 𝑡𝐸) , 𝐻𝑚 = 1 𝐺 𝑓𝑇𝑚 (𝑡) = (𝐺−1)!2 𝐷 1 1 1 (3) 2 𝜒𝑁 , 𝐻𝑚 = 0
Fig. 1: Block diagram of ranked square law selection detection (R-SLS) scheme.
Algorithm 1 Ranked square law selection (R-SLS) with fixed threshold 𝜏 1: Select maximum energy for each of the 𝐿 channels from all 𝑃 diverity branches. 2: Sort the selected channels in ascending order with regard to 𝑇𝑚 . 3: Make decisions for first 𝐿𝑑 channels, where 𝑆𝑑 ≤ 𝐿𝑑 ≤ 𝐿, using energy detection with the given threshold 𝜏 . Ignore the remaining 𝑀 − 𝐿𝑑 channels.
where 𝜒2𝑁 is the chi-square pdf with 𝑁 degrees of freedom, 𝑚 𝐷 = (1 + 𝛾¯ /𝑚1 ) 1 , 𝐸 = 0.5 − 0.5𝑚1 /(𝑚1 + 𝛾¯ ), 𝛾¯ is the average SNR, 𝐺 = 𝑁/2, 𝑀 = 𝐺 − 1, 𝑚1 is the Nakagami parameter and 1 𝐹1 (⋅; ⋅; ⋅) is the confluent hypergeometric function [20]. Since we have 𝐿 = 𝑃 = 2, we first select maximum energy 1 2 , 𝑇𝑚 }, 𝑚 = 1, 2 represents 𝑇𝑚𝑎𝑥𝑚 i.e., 𝑇𝑚𝑎𝑥𝑚 = 𝑚𝑎𝑥{𝑇𝑚 the index of the channel and the superscripts 1 and 2 represent indices of the diversity branches. From this the pdf of 𝑇𝑚𝑎𝑥𝑚 is obtained as [21], 𝑓𝑇𝑚𝑎𝑥𝑚 (𝑡) = 𝑓𝑇𝑚𝑎𝑥 (𝑡) = 2𝐹𝑇𝑚 (𝑡)𝑓𝑇𝑚 (𝑡),
(4)
where, 𝐹𝑇𝑚 (𝑡) represents the cumulative distribution function (CDF) of 𝑇𝑚 . Note that 𝑓𝑇𝑚𝑎𝑥𝑚 (𝑡) is the same for 𝑚 = 1, 2, since it represents pdf of the received energy. Using series representation of confluent hypergeometric function given by ∑∞ (𝑎)𝑘 𝑧𝑘 Γ(𝑎+𝑘) 1 𝐹1 (𝑎, 𝑏, 𝑧) = 𝑘=0 (𝑏)𝑘 𝑘! , where, (𝑎)∫𝑘 = Γ(𝑎) represent∞ 𝑧−1 −𝑥 ing the Pochhammer symbol, Γ(𝑧) = 0 𝑥 𝑒 𝑑𝑥 being
the Gamma function and using Eq. (3)-(4), one can obtain 𝑓𝑇𝑚𝑎𝑥𝑚 (𝑡) = 𝑓𝑇𝑚𝑎𝑥 (𝑡) as ⎧ ∞ ∑ 𝑡𝑀 (𝑚1 )𝑘 1 𝐹1 (𝑚1 , 𝐺, 𝑡𝐸)𝛾(𝐺 + 𝑘, 𝑡 ) 2 ⎨ , 𝐻𝑚 = 1 𝑡 𝑓𝑇𝑚𝑎𝑥 (𝑡) = 𝑘=0 𝑒 2 (𝐺)𝑘 𝐸 −𝑘 Γ(𝐺)2 2𝐺−𝑘−1 𝐷2 𝑘! ⎩2𝜒2 𝑃 (𝐺, 𝑡 ), 𝐻 =0 𝑁
Now, PS in this scenario can be obtained as ∫
𝑃 𝑆01 (𝜏 ) =
𝑃 𝑆(𝜏 ∣𝐻1 = 0, 𝐻2 = 0) = 𝑃 𝑆00 (𝜏 ) = 𝑃 𝑟{𝑇𝑚𝑖𝑛 < 𝜏 },
(6)
where, } 𝑇𝑚𝑎𝑥1 = { 𝑇𝑚𝑖𝑛} = 𝑚𝑖𝑛{𝑇𝑚𝑎𝑥1 , 𝑇𝑚𝑎𝑥{2 }. Here, 𝑚𝑎𝑥 𝑇11 , 𝑇12 and 𝑇𝑚𝑎𝑥2 = 𝑚𝑎𝑥 𝑇21 , 𝑇22 . Considering 𝑇𝑚𝑎𝑥1 and 𝑇𝑚𝑎𝑥2 to be independent and identically distributed, the pdf of 𝑇𝑚𝑖𝑛 can be obtained as [21], 𝑓𝑇𝑚𝑖𝑛 (𝑡) = 2𝑓𝑇𝑚𝑎𝑥 (𝑡) − 2𝐹𝑇𝑚𝑎𝑥 (𝑡)𝑓𝑇𝑚𝑎𝑥 (𝑡),
(7)
where, 𝐹𝑇𝑚𝑎𝑥 (𝑡) is the CDF of 𝑇𝑚𝑎𝑥 . Using 𝑓𝑇𝑚𝑎𝑥 (𝑡) from Eq. (5) under 𝐻𝑚 = 0, the pdf 𝑓𝑇𝑚𝑖𝑛 (𝑡) when both the channels are OFF can be written as 𝑡
×
{𝑀 +𝑙 ∑ 𝑣=0
−
{ 𝑀∑ +𝑙+ℎ 𝑣=0
×
𝛾(𝑎, 𝑧) = (𝑎 − 1)! 1 − 𝑒
𝑎−1 ∑
𝑧𝑘 𝑘!
𝑡1
) 𝑓𝑇𝑚𝑎𝑥2 (𝑡2 )𝑑𝑡2 𝑑𝑡1 .
(12)
Γ(𝑚1 + 𝑘)𝐸 𝑘+𝑙 (𝑚1 )𝑙 Γ(𝐺 + 𝑙) Γ(𝐺)2 𝐷2 2𝐺−𝑘−𝑙−2 𝑘! 𝑙! Γ(𝑚1 )(𝐺)𝑙
} 𝑀 +𝑙 ∑ 𝑀 ∑ 𝛾(𝐺 + 𝑣 + 𝑞, 3𝜏 )2𝐺 𝛾(𝐺 + 𝑣, 𝜏 ) 2 − 2𝑣 𝑣! 𝑣! 𝑞! 3𝐺+𝑣+𝑞 𝑣=0 𝑞=0
𝑘,𝑙=0 ℎ=0
Γ(𝑚1 + 𝑘)𝐸 𝑘+𝑙 (𝑚1 )𝑙 Γ(𝐺 + 𝑙 + ℎ) . (𝐺)𝑙 𝐷2 𝑙! ℎ! 𝑘! Γ(𝐺)2 Γ(𝑚1 )22𝐺+ℎ−𝑘−2
(13)
Similarly, the probability of interference opportunity (PI) can be written as 𝑃 𝐼(𝜏 ∣𝐻1 = 0, 𝐻2 = 1) = 𝑃 𝐼01 (𝜏 ) = 𝑃 𝑟{𝑇𝑚𝑎𝑥2 < 𝑇𝑚𝑎𝑥1 , 𝑇𝑚𝑎𝑥2 < 𝜏 }. (14)
This can be obtained as ∫
𝑃 𝐼01 (𝜏 ) =
𝜏 0
(
𝑓𝑇𝑚𝑎𝑥2 (𝑡2 )
∫
∞ 𝑡2
) 𝑓𝑇𝑚𝑎𝑥1 (𝑡1 )𝑑𝑡1 𝑑𝑡2
(15)
Following a similar procedure as done for 𝑃 𝑆01 (𝜏 ), we can show that 𝑃 𝐼01 (𝜏 ) =
∞ ∑ 𝑀 ∑ (𝑚1 )𝑘 (𝑚1 )𝑙 𝐸 𝑘+𝑙 𝛾(𝐺 + 𝑙 + 𝑞, 𝜏 ) (𝐺)𝑙 Γ(𝐺)𝐷2 𝑘! 𝑙! 𝑞! 2𝐺+𝑞−𝑘−2 𝑞=0
𝑘,𝑙=0
𝑘,𝑙=0
− { ×
)
+𝑣 𝑀 𝑀 ∞ ∑ ∑ ∑ (𝑚1 )𝑘 (𝑚1 )𝑙 𝐸 𝑘+𝑙 Γ(𝐺 + 𝑣) (𝐺)𝑙 Γ(𝐺)2 𝐷2 𝑘! 𝑙! ℎ! 𝑣! 𝑣=0
𝑘,𝑙=0
2
3𝜏 2
ℎ=0
(8)
}
.
∞
+𝑘 ∞ ∑ 𝑀 𝑀 ∑ ∑ (𝑚1 )𝑘 (𝑚1 )𝑙 𝐸 𝑘+𝑙 2𝑙+𝑘+2 𝛾(𝐺 + 𝑙 + 𝑞 + ℎ, − (𝐺)𝑙 Γ(𝐺)𝐷2 𝑘! 𝑙! 𝑞! ℎ! 3𝐺+𝑙+𝑞+ℎ 𝑞=0
Now using the series representation of lower incomplete Gamma function 𝛾(𝑎, 𝑧) for integer value of 𝑎 given by −𝑧
𝑘,𝑙=0
+𝑘 ∞ 𝑀 ∑ ∑
𝑘=0
{
∫
𝑓𝑇𝑚𝑎𝑥1 (𝑡1 )
} 𝑀∑ +𝑙+ℎ ∑ 𝑀 𝛾(𝐺 + 𝑣, 3𝜏 )2𝐺+𝑣 𝛾(𝐺 + 𝑣 + 𝑞, 2𝜏 ) 2 − 𝑣! 3𝐺+𝑣 𝑣! 𝑞! 2𝐺+2𝑞+𝑣 𝑣=0 𝑞=0
𝑡
𝛾(𝐺, 2𝑡 )𝑡𝑀 𝑒− 2 𝛾(𝐺, 2𝑡 )2 𝑡𝑀 𝑒− 2 𝑓𝑇𝑚𝑖𝑛 (𝑡) = − Γ(𝐺)2 2𝐺−2 Γ(𝐺)3 2𝐺−3 𝑀 ∑ 𝛾(𝐺, 𝑡 )𝑡𝑀 𝑒− 2𝑡 𝛾(𝐺 + 𝑘, 𝑡) 2 + . Γ(𝐺)3 2𝑁 +𝑘−3 𝑘!
∞ ∑
𝑃 𝑆01 (𝜏 ) =
(5)
∫ 𝑧 𝑎−1 where, 𝛾(𝑎, 𝑧) = 0 𝑥 𝑒𝑧 𝑑𝑥 is the lower incomplete Gamma function and 𝑃 (𝑎, 𝑧) = 𝛾(𝑎,𝑧) Γ(𝑎) is the normalized incomplete Gamma function. The pdf obtained in Eq. (5) under 𝐻𝑚 = 1 is in the form of infinite series. One can show that this series converges, the proof of which is given in APPENDIX A. Now, there are three possible cases for occupancy of the two channels 𝐶1 and 𝐶2 . These correspond to: 1. Both the channels are OFF, 2. Only one channel is ON and 3. Both the channels are ON. If both the channels are OFF, one can write the probability of spectrum opportunity (PS) as
0
(
Using the pdfs given in Eq. (5), 𝑃 𝑆01 (𝜏 ) can be shown to be
𝑚
2
𝜏
ℎ=0
} 𝑀 +𝑘 ∑ 𝛾(𝐺 + 𝑙 + ℎ, 3𝜏 ) 𝛾(𝐺 + 𝑙 + 𝑞 + ℎ, 2𝜏 ) 2 − 22𝐺+𝑣 1.5𝐺+𝑙+ℎ 𝑞! 23𝐺+𝑣+𝑙+2𝑞+ℎ−𝑘−2 𝑞=0
𝑘+2
(9)
(16)
Note that, one can obtain the same expressions for PS and PI as in Eq. (13) and Eq. (16), respectively, assuming channel 1 PS when both channels are OFF can be obtained by integrating as ON and channel 2 as OFF, i.e., 𝑃 𝑆01 (𝜏 ) = 𝑃 𝑆10 (𝜏 ) and 𝑓𝑇𝑚𝑖𝑛 (𝑡) from 0 to 𝜏 . This can be shown to be 𝑃 𝐼01 (𝜏 ) = 𝑃 𝐼10 (𝜏 ). 𝑀 𝑀 𝜏 𝜏 Now, if both channels are ON, we have 𝑃 𝑆(𝜏 ∣𝐻1 = ∑ ∑ 4𝛾(𝐺, Γ(𝐺 + 𝑘)𝛾(𝐺, ) ) 12𝛾(𝐺 + 𝑙, 𝜏 ) 2 2 𝑃 𝑆00 (𝜏 ) = + − 1, 𝐻 𝐺+𝑙 2 𝐺+𝑘−3 2 = 1) = 𝑃 𝑆11 (𝜏 ) = 0, as this results in zero spectrum Γ(𝐺)2 𝑙! Γ(𝐺) 2 𝑘! Γ(𝐺) 𝑙=0 𝑘=0 opportunities. The PI in this case can be written as } { 𝑀 𝑀 +𝑘 𝑀 𝑘=0
−
∑ ∑ 23−𝑘 Γ(𝐺 + 𝑘) 𝛾(𝐺 + ℎ, 3𝜏 ) ∑ 𝛾(𝐺 + 𝑙 + ℎ, 2𝜏 ) 2 − Γ(𝐺)2 ℎ! 𝑘! 2𝐺 2−ℎ 3𝐺+ℎ 2𝑁 +2𝑙+ℎ 𝑙! 𝑘=0 ℎ=0 𝑙=0 } { 𝑀 ∑ 8𝛾(𝐺 + 𝑙 + ℎ, 3𝜏 ) Γ(𝐺 + 𝑘)𝛾(𝐺 + 𝑙, 𝜏 ) 2 − + (10) 3𝐺+𝑙+ℎ Γ(𝐺)𝑙! ℎ! Γ(𝐺)2 2𝑁 +𝑙+𝑘−3 𝑙! 𝑘!
𝑃 𝐼(𝜏 ∣𝐻1 = 1, 𝐻2 = 1) = 𝑃 𝐼11 (𝜏 ) = 𝑃 𝑟{𝑇𝑚𝑖𝑛 < 𝜏 }. (17) Using pdf under 𝐻𝑚 = 1 from Eq. (5) and using Eq. (7), the pdf of 𝑇𝑚𝑖𝑛 can be obtained as 𝑡 ∞ ∑ 𝑡𝐺−1 𝑒− 2 1 𝐹1 (𝑚1 , 𝐺, 𝑡𝐸)𝛾(𝐺 + 𝑘, 2𝑡 ) 𝑓𝑇𝑚𝑖𝑛 (𝑡) = Γ(𝐺)2 𝐷2 2𝐺−𝑘−2 Γ(𝐺 + 𝑘)Γ(𝑚1 )
𝑙,ℎ=0
Since both the channels are OFF, there is no chance of interference and hence, 𝑃 𝐼(𝜏 ∣𝐻1 = 0, 𝐻2 = 0) = 𝑃 𝐼00 (𝜏 ) = 0. Next, we consider the case when only one channel is ON. Assuming channel 1 is OFF and channel 2 is ON, PS can be written as
𝑘=0
−
∞ ∑ 𝑡𝑀 1 𝐹1 (𝑚1 , 𝐺, 𝑡𝐸)Γ(𝑚1 + 𝑟)𝐸 𝑟+𝑘 𝛾(𝐺 + 𝑟, 2𝑡 )Γ(𝑚1 + 𝑘) 𝑡
𝑟,𝑘=0
𝑃 𝑆(𝜏 ∣𝐻1 = 0, 𝐻2 = 1) = 𝑃 𝑆01 (𝜏 ) = 𝑃 𝑟{𝑇𝑚𝑎𝑥1 < 𝑇𝑚𝑎𝑥2 , 𝑇𝑚𝑎𝑥1 < 𝜏 }. (11)
𝑒 2 Γ(𝐺)2 𝐷4 22𝐺−𝑘−𝑟−3 Γ(𝐺 + 𝑟)Γ(𝑚1 )2 𝑘! 𝑟! { ∞ }} 𝑀 +𝑘 ∑ (𝑚1 )𝑙 𝐸 𝑙 { 𝐺+𝑙 ∑ 𝛾(𝐺 + 𝑙 + ℎ, 𝑡) 𝑡 𝛾(𝐺 + 𝑙, ) − 2 . (𝐺)𝑙 𝑙! 2 ℎ! 2ℎ 𝑙=0 ℎ=0 (18)
Using Eq. (18), 𝑃 𝐼 in this case can be obtained by integrating 𝑓𝑇𝑚𝑖𝑛 (𝑡) from 0 to 𝜏 and is given by Eq. (19) shown on the top of next page. Note that few of the equations obtained here are in terms of infinite series. Sufficient accuracy can be obtained by truncating them to finite number of terms. Using 𝑃 𝑆𝑖𝑗 (𝜏 ) and 𝑃 𝐼𝑖𝑗 (𝜏 ), for 𝑖, 𝑗 = 0, 1, probability of a spectrum opportunity (𝑃 𝑆(𝜏 )) and the probability of interference opportunity (𝑃 𝐼(𝜏 )) for the selected threshold 𝜏 can be computed using 𝑃 𝑆(𝜏 ) = (1−𝑝)2 𝑃 𝑆00 (𝜏 )+2𝑝(1−𝑝)𝑃 𝑆01 (𝜏 )+𝑝2 𝑃 𝑆11 (𝜏 ), (20)
and 𝑃 𝐼(𝜏 ) = (1 − 𝑝)2 𝑃 𝐼00 (𝜏 ) + 2𝑝(1 − 𝑝)𝑃 𝐼01 (𝜏 ) + 𝑝2 𝑃 𝐼11 (𝜏 ). (21)
Finally, using Eq. (20) and Eq. (21), the probability of insufficient spectrum opportunity and the probability of excessive interference opportunity can be obtained as 𝑃𝐼𝑆𝑂 (𝜏 ) = 𝑃 𝑟{𝑆 < 𝑆𝑑 } = 1 − 𝑃 𝑆(𝜏 ),
(22)
𝑃𝐸𝐼𝑂 (𝜏 ) = 𝑃 𝑟{𝐼 > 𝐼𝑑 } = 𝑃 𝐼(𝜏 ),
(23)
respectively. V. R ESULTS AND D ISCUSSION In order to study the performance of proposed R-SLS scheme we discuss analytical results using equations derived in section IV. The simulations are also carried out to study the effect of different parameters on the performance of R-SLS. The performance is measured by using various plots of 𝑃𝐸𝐼𝑂 Vs 𝑃𝐼𝑆𝑂 under Nakagami fading channel. Note that plots in Fig. (3)-(6) are obtained using Monte-Carlo simulations. In Fig. 2 (a), we display 𝑃𝐸𝐼𝑂 Vs 𝑃𝐼𝑆𝑂 for ranked channel detection (RCD) and the proposed R-SLS. The plot for RCD is obtained using 𝑚1 = 2, 𝐿 = 2, 𝐿𝑑 = 𝑆𝑑 = 1, 𝐼𝑑 = 0, 𝑝 = 0.1, 𝑁 = 10 and 𝛾¯ = 0 𝑑𝐵. The plot for R-SLS is obtained using the same parameters as RCD considering two diversity branches, i.e., 𝑃 = 2. The plot for R-SLS is obtained using Eq. (10), Eq. (13), Eq. (16), Eq. (19) and Eq. (20)-(23). Note that, Eq. (13), Eq. (16) and Eq. (19) are in the form of infinite series. The convergence of these series can be proved following a similar procedure as given in APPENDIX A. However, sufficient accuracy can be obtained by taking finite terms in these infinite series expressions. For example, with 𝑁 = 10, 𝛾¯ = 0 𝑑𝐵, 𝑚1 = 2 and 𝜏 = 10, accuracy of upto five decimal points can be obtained for the series in Eq. (13) by taking upper limit as 12 for both 𝑘 and 𝑙. Looking at the plots in Fig. 2 (a), we observe that for 𝑃𝐸𝐼𝑂 ≈ 0.04 the corresponding 𝑃𝐼𝑆𝑂 is approximately 0.45 for RCD and it is 0.35 for R-SLS indicating the improvement in spectrum opportunity using the proposed detection algorithm. In section IV we considered a simple case of 𝐿 = 𝑃 = 2 to avoid mathematical rigor. Hence, we also show Monte-Carlo simulations to study the performance using general parameter settings. Fig. 2 (b), shows the plots using equations and Monte-Carlo simulation for R-SLS using the same parameters. The parameters are kept same as in Fig. 2(a) for R-SLS. The overlapping of the two
plots indicates correctness of simulation. For simulations we used BPSK primary signal with Nakagami fading channel and the results are averaged over 105 realizations. Fig. 3 (a), shows 𝑃𝐸𝐼𝑂 Vs 𝑃𝐼𝑆𝑂 for the R-SLS considering 𝑁 = 10, 𝐿 = 8, 𝑃 = 4, 𝑆𝑑 = 1, 𝐼𝑑 = 0, 𝐿𝑑 = 2, 𝑝 = 0.1 for different values of 𝛾¯ ranging from −10 𝑑𝐵 to 0 𝑑𝐵. It is observed that, with increase in 𝛾¯ , the detection performance improves. In Fig. 3 (b), we demonstrate the effect of occupancy probability 𝑝 on the performance. For this we consider same parameter settings as in Fig. 3 (a) with 𝛾¯ = 0 𝑑𝐵 and 𝑝 is varied from 0.05 to 0.3. We can see that as 𝑝 increases the number of ON channels in the partial band increases and this reduces the chances of getting the free channels and increases chances of interference. This can be observed in the plot of Fig 3 (b) where we see that the performance is better for 𝑝 = 0.05 and it degrades as 𝑝 increases. Fig. 4 (a) shows the simulation results for the proposed RSLS with increase in the number of diversity branches. Here, the number of branches 𝑃 is varied from 1 to 8. It is observed that performance is enhanced as the number of branches is increased. Note that, 𝑃 = 1 corresponds to no diversity case, i.e., it corresponds to RCD algorithm proposed in [6]. In Fig. 4(b) we demonstrate the effect of increasing the number of channels in the partial band 𝐿 while keeping the number of diversity branches 𝑃 fixed at 4. We varied 𝐿 from 4 to 16 and found that increasing 𝐿 results in improved spectrum opportunity. However, this increases the sampling rate. In Fig. 5(a), plots are obtained for different values of desired number of free channels 𝑆𝑑 , keeping 𝐿𝑑 = 4 and the other parameters same as in Fig. 4. It can be seen that as the required number of free channels is increased the performance degrades. Fig. 5(b) shows the plots for different 𝐿𝑑 while keeping 𝑆𝑑 = 1. It is found that increasing 𝐿𝑑 results in performance degradation. This is because though it results in spectrum opportunities it also increases the number of interference opportunities. Hence, it is advantageous to keep 𝑆𝑑 = 𝐿𝑑 to minimize the interference to the primary users. Fig. (6) displays the effect of increasing allowed number of interference to the primary users, i.e., 𝐼𝑑 . The simulation is carried out using 𝑁 = 10, 𝑝 = 0.5, 𝐿 = 16, 𝑃 = 4, 𝛾¯ = 0 𝑑𝐵, 𝑚1 = 2, 𝐿𝑑 = 8, 𝑆𝑑 = 4 and 𝐼𝑑 is varied from 0 to 2. It is found that increase in 𝐼𝑑 results in more spectrum opportunities to secondary users. This is because using 𝐼𝑑 > 0 makes secondary users to interfere with primary users. VI. C ONCLUSION In this paper we studied the problem of wideband spectrum sensing for cognitive radio under Nakagami fading channel. We propose ranked square low selection detection algorithm which uses diversity reception. We provide complete mathematical analysis for the proposed algorithm. We found that using diversity, the detection performance improves. We observe that ranked square law selection algorithm outperforms ranked channel detection when used without diversity. We also study the effect of different parameters on the performance using simulations.
} 𝑀 +𝑟 ∞ ∑ ∑ 8(𝑚1 )𝑘 (𝑚1 )𝑟 (𝑚1 )𝑙 (𝑚1 )𝑞 𝑃 (𝐺 + 𝑞, 𝜏2 ) 𝛾(𝐺 + 𝑘 + 𝑙, 𝜏 ) 𝜏 2 𝛾(𝐺 + 𝑘, ) − − 2 2𝐺+𝑙 𝑙! 𝐷4 𝑟! 𝑘! 𝑙! 𝑞! (2𝐸)−𝑟−𝑘−𝑙−𝑞 𝑟,𝑘=0 𝑙=0 𝑟,𝑘,𝑙,𝑞=0 } { 𝐺+𝑙−1 𝑀 +𝑟 ∞ ∑ ∑ ∑ 2𝑞 𝛾(𝐺 + 𝑣 + 𝑞 + 𝑠, 3𝜏 ) (𝑚1 )𝑟 (𝑚1 )𝑘 (𝑚1 )𝑙 (𝑚1 )𝑞 𝐸 𝑟+𝑘+𝑙+𝑞 2𝑟+𝑘+𝑙+3 𝛾(𝐺 + 𝑣 + 𝑞, 𝜏 ) 2 + − (𝐺)𝑞 𝐷4 Γ(𝐺)𝑟! 𝑘! 𝑙! 𝑞! 𝑣! 2𝐺+𝑣−1 𝑠! 3𝐺+𝑣+𝑞+𝑠 𝑠=0 𝑟,𝑘,𝑙,𝑞=0 𝑣=0 { } 𝑀 +𝑟 𝑀 +𝑘 ∞ ∑ ∑ ∑ (𝑚1 )𝑟 (𝑚1 )𝑘 (𝑚1 )𝑙 (𝑚1 )𝑞 Γ(𝐺 + 𝑙 + ℎ)𝐸 𝑟+𝑘+𝑙+𝑞 𝛾(𝐺 + 𝑣 + 𝑞, 𝜏 ) 𝜏 𝑞 2 𝛾(𝐺 + 𝑞, ) − + (𝐺)𝑞 (𝐺)𝑙 Γ(𝐺)2 𝐷4 𝑟! 𝑘! 𝑙! 𝑞! ℎ! 2𝐺+ℎ−𝑟−𝑘−3 2 2𝐺+𝑣 𝑣! 𝑣=0 𝑟,𝑘,𝑙,𝑞=0 ℎ=0 { } 𝑀 +𝑟 𝑀 +𝑘 𝑀∑ +𝑙+ℎ ∞ ∑ ∑ ∑ ) (𝑚1 )𝑟 (𝑚1 )𝑘 (𝑚1 )𝑙 (𝑚1 )𝑞 Γ(𝐺 + 𝑙 + ℎ)𝐸 𝑟+𝑘+𝑙+𝑞 𝛾(𝐺 + 𝑠 + 𝑞, 3𝜏 𝛾(𝐺 + 𝑠 + 𝑣 + 𝑞, 2𝜏 ) 2 − (19) (𝐺)𝑙 (𝐺)𝑞 𝐷4 Γ(𝐺)2 𝑟! 𝑘! 𝑙! 𝑞! ℎ! 𝑠! 2𝐺+ℎ−𝑟−𝑘−3 2−𝑠−𝑞 3𝐺+𝑠+𝑞 22𝐺+2𝑣+𝑞+𝑠 𝑣! 𝑠=0 𝑣=0
∞ ∑ (𝑚1 )𝑟 (𝑚1 )𝑘 𝐸 𝑟+𝑘 2𝑟+2 𝑃 𝐼11 (𝜏 ) = 𝐷2 𝑟! 𝑘! Γ(𝐺 + 𝑘)
−
{
𝑘
𝑟,𝑘,𝑙,𝑞=0 ℎ=0
100
100 Theoretical Simulation
P ISO
P ISO
P ISO
L=4 L=8 L=16
P=1 P=2 P=4 P=8
10-1
10-1
10-2
10-2
10-1
10-1
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
P EIO
P EIO
(a)
0.05
(b)
0.15
0.02
0.04
0.06
0.08
0.1
0.12
(b)
Fig. 4: 𝑃𝐸𝐼𝑂 versus 𝑃𝐼𝑆𝑂 for R-SLS using 𝑁 = 10, 𝑝 = 0.1, 𝑆𝑑 = 1, 𝐼𝑑 = 0, 𝐿𝑑 = 2, 𝛾¯ = 0 𝑑𝐵, 𝑚1 = 2. (a) plots for 𝐿 = 8 with different number of diversity branches (𝑃 ). (b) plots for 𝑃 = 4 with different number of channels in the partial band (𝐿).
100
100
100
p=0.05 p=0.10 p=0.20 p=0.30
S =1 d
Ld = 1
S =2
Ld = 2
d
S =3 d
P ISO
P ISO
-10 dB -5 dB 0 dB
10-1
0
(a)
Fig. 2: (a) 𝑃𝐸𝐼𝑂 versus 𝑃𝐼𝑆𝑂 showing comparison of RCD and R-SLS detection algorithms using 𝑁 = 10, 𝐿 = 2, 𝑆𝑑 = 1, 𝐼𝑑 = 0, 𝐿𝑑 = 1, 𝑝 = 0.1, 𝛾¯ = 0 𝑑𝐵, 𝑚1 = 2 for RCD and 𝑃 = 2 for R-SLS with other parameters same as RCD. (b) validity of Monte-Carlo simulation for R-SLS. The parameters used are same as that of part (a) for R-SLS.
100
0.1
P EIO
P ISO
0
P ISO
100
100
RCD R-SLS
10-1 10-1 10-1
10-2
0
0.05
0.1
P EIO
(a)
0.15
0.2
0
0.05
0.1
0.15
0.2
0.25
0.3
P EIO
(b)
Fig. 3: 𝑃𝐸𝐼𝑂 versus 𝑃𝐼𝑆𝑂 for R-SLS using 𝑁 = 10, 𝐿 = 8, 𝑃 = 4, 𝑆𝑑 = 1, 𝐼𝑑 = 0, 𝐿𝑑 = 2, 𝑚1 = 2. (a) plots for 𝑝 = 0.1 with different values of 𝛾¯ . (b) plots for different 𝑝 with 𝛾¯ = 0 𝑑𝐵.
0
0.05
0.1
0.15
P EIO
(a)
0.2
0.25
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
P EIO
(b)
Fig. 5: 𝑃𝐸𝐼𝑂 versus 𝑃𝐼𝑆𝑂 for R-SLS using 𝑁 = 10, 𝑝 = 0.1, 𝐿 = 8, 𝑃 = 4, 𝐼𝑑 = 0, 𝛾¯ = 0 𝑑𝐵, 𝑚1 = 2. (a) plots for 𝐿𝑑 = 4 with different values of 𝑆𝑑 . (b) plots for 𝑆𝑑 = 1 with different values of 𝐿𝑑 .
R EFERENCES
1 0.9 0.8 0.7 0.6
P ISO
0.5 0.4 0.3 I =0 d
I =1 d
0.2
I d =2
0
0.2
0.4
0.6
0.8
1
P EIO
Fig. 6: 𝑃𝐸𝐼𝑂 versus 𝑃𝐼𝑆𝑂 for R-SLS using 𝑁 = 10, 𝑝 = 0.5, 𝐿 = 16, 𝑃 = 4, 𝛾¯ = 0 𝑑𝐵, 𝑚1 = 2, 𝐿𝑑 = 8, 𝑆𝑑 = 4 for different number of 𝐼𝑑 . A PPENDIX A P ROOF FOR CONVERGENCE OF 𝑓𝑇𝑚𝑎𝑥 (𝑡) The convergence of infinite series in Eq.(6) can be proved using the ratio test which is given by 𝐿𝑟𝑎𝑡𝑖𝑜
𝐶𝑘+1 . = lim 𝑘→∞ 𝐶𝑘
(24)
Here, 𝐶𝑘+1 and 𝐶𝑘 are the (𝑘 + 1)𝑡ℎ and 𝑘 𝑡ℎ terms of the series, respectively. Using Eq.(24), one can test the convergence as follows ∙ If 𝐿𝑟𝑎𝑡𝑖𝑜 < 1, the series is convergent. ∙ If 𝐿𝑟𝑎𝑡𝑖𝑜 < 1, the series is divergent. ∙ If 𝐿𝑟𝑎𝑡𝑖𝑜 = 1, the series may be divergent, conditionally convergent, or absolutely convergent. Now, using the series in Eq.(6), 𝐿𝑟𝑎𝑡𝑖𝑜 can be obtained as 2𝐸(𝑚1 + 𝑘)𝑃 (𝐺 + 𝑘 + 1, 2𝑡 ) . 𝐿𝑟𝑎𝑡𝑖𝑜 = lim 𝑡 𝑘→∞ (𝑘 + 1)𝑃 (𝐺 + 𝑘, )
(25)
2
Using the recurrence formula for normalized incomplete Gamma function given in [22, (6.5.21)], i.e., 𝑃 (𝑎 + 1, 𝑥) = 𝑥𝑎 𝑒−𝑥 , 𝐿𝑟𝑎𝑡𝑖𝑜 can be written as 𝑃 (𝑎, 𝑥) − Γ(𝑎+1) 𝐿𝑟𝑎𝑡𝑖𝑜
} { 𝑡 ( 2𝑡 )𝐺+𝑘 𝑒− 2 2𝐸(𝑚1 + 𝑘) = lim 1− (26) 𝑡 𝑘→∞ (𝑘 + 1) (𝐺 + 𝑘)𝛾(𝐺 + 𝑘, 2 )
Using the series representation of lower incomplete Gamma 𝑙 𝑎+𝑙 ∑∞ 𝑥 function as 𝛾(𝑎, 𝑥) ≈ 𝑙=0 (−1) and taking limit 𝑘 → (𝑎+𝑙)𝑙! ∞, we can show that 𝐿𝑟𝑎𝑡𝑖𝑜
⎧ ⎫ ⎨ 𝑡 ⎬ −2 𝑒 = 2𝐸 1 − ∑ 𝑡 ∞ (−1)𝑙 ( 2 )𝑙 ⎭ ⎩ 𝑙=0 𝑙!
Using the fact that 𝐿𝑟𝑎𝑡𝑖𝑜 as
∑∞
𝑙=0
(−1)𝑙 ( 2𝑡 )𝑙 𝑙!
(27)
𝑡
= 𝑒− 2 , we can obtain
𝐶𝑘+1 = 0. 𝐿𝑟𝑎𝑡𝑖𝑜 = lim 𝑘→∞ 𝐶𝑘
(28)
Since, 𝐿𝑟𝑎𝑡𝑖𝑜 in Eq.(31) is
