approaches in this work. The derived frameworks are based on the canonical series representation of Marcum. Q-function with derivatives of moment generating.
On the Energy Detection of Unknown Deterministic Signals over Generalized Fading Channel O. Olabiyi, S. Alam, O. Odejide, A. Annamalai, Center of Excellence for Communication Systems Technology Research Department of Electrical and Computer Engineering Prairie View A&M University, TX 77446 Abstract— In this article, we develop new analytical techniques for the performance analysis of energy detection with selection diversity combining (SDC) and square-law selection (SLS) diversity schemes over generalized fading channels. We present two novel approaches in this work. The derived frameworks are based on the canonical series representation of Marcum Q-function with derivatives of moment generating function (MGF) of signal-to-noise ratio (SNR) of fading channel and single integral of cumulative distribution function (CDF) of SNR for SLS and SDC diversity schemes respectively. Using these frameworks, we found out that the performance of both SLS and SDC diversity schemes improves with smaller number of samples, increasing fading index, mean SNR and number of diversity branches. Also, comparing both diversity schemes shows that the choice of better detection diversity scheme varies with mean SNR. At low SNR, SDC performs better than the SLC scheme, but as the mean SNR increases, SLS gives better performance while high mean SNR values favours again the SDC scheme. More so, lower number of samples favours the SLS scheme than the SDC scheme. Although, the framework is applicable across fading channels, selected numerical results are presented on the Rice fading channel since there are limited results in literature in this area. Keywords— Energy detection, Cognitive radio, Fading channels, Selection diversity combining, Square-law-selection
1 Introduction The emerging technology of cognitive radio has created a paradigm shift in the design of wireless system where radio can now adapt their operating behaviour to take advantage of unused spectrum. One of the main requirements of this system is to ensure that the incumbent (i.e. primary or licensed user) is not interrupted by the activity of these cognitive radios. Therefore the new radio must be capable of determining the presence or absence of an incumbent before spectrum usage. Among the various known spectrum sensing techniques, blind sensing using energy detectors is perhaps the simplest (i.e., low complexity) and more importantly, the “secondary users” do not require any unauthorized details or a priori knowledge of the “primary user” transmissions (i.e., the energy detector
of a secondary user treats the received primary user transmission as an unknown deterministic signal). While energy detection of the received signal waveform over an observation time window may be more versatile than the cyclo-stationary feature detection approach, its performance (reliability) is severely limited by multipath fading. Therefore, performance characterization of energy detectors with diversity reception over different wireless fading environments is of practical interest. Urkowitz, [1] first studied the detection of an unknown deterministic signal over a flat band-limited Gaussian noise channel using an energy detector. The results in [1] are further extended to Rayleigh, Rice and Nakagami fading channels in [2]. Average probabilities of detection (Pd) and false alarm probability (Pf) over Rayleigh, Rice and Nakagami-m fading channels are presented. The probability of detection was derived in closed-form for Rayleigh fading channel; however the Nakagami fading channels results to single integral solution while the Rice fading channel’s result is an infinite summation. Nakagami-m and Rice fading channels are considered in [3] and [4]. The reported result for single Nakagami-m channel is limited to integer values of the shape parameter (m) whereas the result for Rice fading channel is restricted to unity time-bandwidth product. The performance of energy detector in diversity system such as Maximal ratio combining (MRC), square-law selection (SLS) and switch and stay (SSC) diversity detectors over i.i.d Rayleigh fading channels was considered in [3] and [4]. In [5]-[7], the Authors analyse the performance of the energy detector with selection diversity combining (SDC) in Nakagami-m channel, MRC in Nakagami-m and Rice channel and equal gain combining (EGC) in Nakagami-m channel respectively with all derived expression in closed form. In [8], performance with SSC, square-law combining (SLC) and SLS were considered. In all these cases, only integer u, integer Nakagami fading index m and integer Lu or Lm (where L is the number of diversity branches) are considered. In [12], we presented using an alternative form of Marcum Q-function a new approach involving derivatives of moment generating function (MGF) of signal-to-noise ratio (SNR) of fading channels and we applied this method to MRC and SLC in Nakagami-m and Rice fading channels. This method easily tackles scenarios with non-integer u, non-integer Nakagami fading index m, Lu or Lm with all expressions involving single infinite summation term.
In this article, we develop novel approaches of analysing the performance of energy detector with SDC and SLS diversity schemes in generalized fading channels. The derived frameworks are based on the canonical series representation of Marcum Q-function with derivatives of MGF of SNR of fading channel and single integral of cumulative distribution function (CDF) of SNR for SLS and SDC diversity schemes respectively. However, simpler expressions were derived for special cases of SDC scheme for both Nakagami-m and Rice fading channels. Using these frameworks, we found out that the performance of both SLS and SDC diversity schemes improves with smaller number of samples, increasing fading index, mean SNR and number of diversity branches. Also, comparing both diversity schemes shows that the choice of better detection diversity scheme varies with mean SNR. Though the derived framework is applicable across fading channels, our numerical result is on Rice fading channel which has not been well researched in literature. The rest of the paper is organized as follows. In section II we present system model, notations and performance over single channel. Section III extends this to SLS diversity system and develops performance analysis for SDC diversity scheme. Finally, numerical results and concluding remarks are presented in section IV and V respectively.
2 System model and notation To be consistent, notations similar to [8] are used as listed below. s (t ) : Unknown deterministic signal waveform
n(t )
: Noise waveform – White Gaussian random process
si
: Unknown deterministic signal waveform
ni
: Noise waveform – White Gaussian random process
r (t )
: Received signal
h T W u = TW N 01 Es
: Channel coefficient amplitude : Observation time interval : One-sided bandwidth : Time-Bandwidth product : One sided noise power spectral density
λ L H0 H1
: Signal energy over the time interval T : Energy threshold of the receiver : Number of branches of the receiver combiner : Hypothesis 0; no s (t ) present : Hypothesis 1; s (t ) present
2 χ 2u
: Central Chi-square distribution with 2u degrees of freedom χ 22u (є ) : Non central Chi-square distribution with 2u degrees of freedom and non centrality parameter є The detection of the existence of the unknown deterministic signal s (t ) by the receiver, is a binary hypothesis test as shown in [1, eq(1)], : H0 n(t ) (1) y (t ) = : H1 hs ( t ) + n ( t )
Therefore, a sample from noise process ni is a Gaussian random variable with zero mean and N 01W variance;
ni ∼ N (0, N 01W ) [1]. The energy detector decision variable Y can be expressed as [1, eq.(2)]. 2 Y= N 01
∫
T
0
n i n (t )dt = ∑ N 01W i =1 2u
2
2
: H0
(2)
Thus, Y under H 0 is a square sum 2u Gaussian random variable of N (0,1) and follows χ 22u . Similarly, Y under H1 is formed as given by (3). 2
hs + n i (3) y (t ) dt = i : H1 0 N W i =1 01 Here, we assume that the channel coefficient amplitude is Y under constant over the 2u samples. It therefore follows that H1 is χ 22u (є j ) where є is given by [1, eq.(4)], 2 Y= N 01
∫
2u
T
∑
2
hs i є= N 01W i =1 2u
∑
2
h2 = N 01W
Here, the SNR is defined by γ =
2u
∑s
2 i
i =1
h 2 Es N 01
=
2 h 2 Es = 2γ N 01
(4)
.
In [12], it has been shown that the detection and false alarm probabilities of an energy detector in AWGN channel is given by:
Pd = Qu ( 2γ , λ )
(5)
and Pf =
Γ (u , λ2 ) Γ (u )
(6)
respectively, where Qu (.,.) is the generalised ( u th order) Marcum-Q-function and Γ(.,.) is the upper incomplete gamma function which is defined by the integral form ∞
Γ ( a, x ) = ∫ t a −1e − t dt and Γ(a, 0) = Γ(a) . Pf is the same over x
any fading channel since there is no γ in (6). In the other sense Pd has to be averaged over different fading channels and diversity combining. Therefore, the average detection probability, Pd is given by ∞
Pd = ∫ Qu ( 2γ , λ ) fγ (γ ) d γ 0
(7)
where fγ (γ ) is the probability density function of SNR, γ . Using the alternative representation of generalized Marcum Q- function, we have shown in [12] that it can be written as, ∞ γ k e−γ G (u + k , λ2 ) (8) Qu ( 2γ , λ ) = 1 − k ! Γ(u + k ) k =0
∑
where G (.,.) is the lower incomplete gamma function which is x
defined by G ( a, x ) = ∫ t a −1e − t dt . 0
Hence, substituting (8) into (6), we obtain the generalized average Pd in this case, Pd Gen as
( −1) k G (u + k , λ2 ) ( k ) φγ ( s ) k ! Γ (u + k ) k =0 ∞
Pd Gen = 1 − ∑
∂ φγ
(9) s =1
and φγ ( s ) is the MGF of SNR of
∂s k different stochastic fading channels. The kth derivatives of the MGF of common fading channels are listed in [12, Table 1]. For example, the final expression for detection probability is easily obtained for Nakagami-m and Rice Fading channels respectively as ∞ 1 G (u + k , λ2 ) Ω k m m Γ(m + k ) Pd Nak = 1 − (10) m+k Γ( m) k = 0 k ! Γ(u + k ) ( m + Ω)
∑
G (u + k , λ2 ) Ω k (k !)(1 + K ) −K Ω exp = 1− 1+ k 1+ K + Ω k = 0 Γ (u + k ) (1 + K + Ω ) ∞
Pd R ic
∑
k
×
∑ i =0
1 K (1 + K ) (i !) 2 (k − 1)! 1 + K + Ω
(11)
i
The convergence and truncation error of this infinite series has been well treated in [12] and it has been shown that only few terms are required for four-digit accuracy. In fact, less than 25 terms are required for no diversity case across different system parameter changes. Next, we will apply this result in deriving the expression for the performance of energy detection over generalized fading channel with postdetection square-law selection (SLS).
3 Detection over generalized fading channel with diversity system
L
Pf SLS = Pr {YSLS > λ | H 0 } = 1 − ∏ Pr {Yl < λ | H 0 } Γ (u , λ2 ) = 1 − 1 − Γ (u ) Assuming independent branch statistics, we obtain,
(13)
L
1 G(u + k , λ2 ) Ωk (k !)2 (1 + Kl ) x Γ(u + k ) (1 + Kl + Ω)1+ k
l =1
k =0
− Kl Ω k Kl (1 + Kl ) 1 exp ∑ 2 1 + K + Ω l i =0 (i !) (k − 1)! 1 + Kl + Ω
(18) i
3.2 Selection diversity combining (SDC) Since the MGF and PDF of i.n.d channels is difficult to obtain in the case of SDC, we abandon the above MGF method to seek a more general solution taking advantage of readily available cumulative distribution function (CDF). In the selection combining detection, the branch with maximum SNR is selected i.e. γ SC = max(γ 1 , γ 2 ...γ L ) (19) Therefore, the CDF of the output SNR of L-branch selection combiner is given by Fγ SC (γ ) =
L
∏F l =1
γl
(γ )
(20)
where Fγ l (γ ) is the CDF of branch l. Since only the samples from a single branch are being considered at any point in time, the decision variable YSC is
SC
l =1
(
= 1 − ∏ 1 − Qu ( 2γ l , λ )
)
(14)
Since there is no γ in (12), Pf SLS is the same across all channels and branches. Averaging (13) over different channels we obtain similar to (8) L ∞ ( −1) k G (u + k , λ ) 2 Pd SLS = 1 − ∏ ∑ φγ(lk ) ( s) (15) k ! Γ (u + k ) l =1 k = 0 s =1 and for i.i.d branches we obtain L
. s =1
χ 2 : H0 YSC = Yl ∼ 2 2u (21) χ 2u (є SC ) : H1 where the non-centrality є SC = єl = 2γ l = 2γ SC , hence the false alarm probability remains the same i.e. Γ(u , λ2 ) (22) Pf = Γ(u ) However, the detection probability is affected due to distribution of the SNR of the combiner’s output and could be written from (7) as ∞
Pd SLS = Pr {YSLS > λ | H1} = 1 − ∏ Pr {Yl < λ | H1}
l
∞
Pd = ∫ Qu ( 2γ , λ ) fγ (γ ) d γ
L
Pd SLS
L
∑ ∏ k!
SC
l =1
∞ (−1)k G (u + k , λ2 ) ( k ) = 1− ∑ φ (s) k = 0 k ! Γ(u + k ) γ
Pd SLS − Ric = 1 −
defined by
Since the branch with maximum output decision is been chosen, the effective decision variable can be written as in [8] YSLS = max(Y1 , Y2 ,..., YL ) (12) Assuming independent branch statistics,
l =1
(17)
l
2 just the i.i.d χ 2u for H 0 and χ 22u (є j ) for H1 and this is
3.1 Square-law selection (SLS)
L
L ∞ 1 G (u + k , λ ) Ω k m m Γ ( m + k ) 2 l l Pd SLS − Nak = 1 − ∏ ∑ m +k Γ(ml ) l =1 k = 0 k ! Γ (u + k ) (ml + Ω) l
k
where φγ( k ) ( s ) =
For example substituting the MGF of Nakagami and Rice channels produces respectively
(16)
0
SC
(23)
where fγ SC (γ ) is the PDF of SNR of the combiner’s output. Our approach here is to first find a generalized solution to (23) and later present simpler solution for special cases. We will like to mention here that only [5] has considered this problem and their result is limited to i.i.d dual combiner (with non integer m) and i.i.d multiple branches with integer m with both cases resulting to solution with combination of infinite summation term and infinite series function [5, Eq. (5), (17), (21)]. In order to obtain amore compact and general solution for (23) that is applicable to i.n.d channels and mixed fading environment, we use the CDF method. Using the integration by part, it is straight forward to show that
∂Qu ( 2γ , λ ) dγ 0 ∂γ Using the identity in [11 eq. (11)], we obtain
∫
PdSC = 1 −
∞
Fγ SC (γ ) u
∞
(24)
λ − (γ + λ2 ) PdSC = 1 − I u 2γλ Fγ SC (γ )d γ (25) e 0 2γ where I u (.) is the u-th order modified Bessel function of the first kind. Substituting (20) into (23), we obtain ∞ L λ − (γ + λ ) Pd = 1 − e I 2 γλ Fγ (γ ) d γ (26) u ∏ 2γ 0
∫
u
∫
SC
(
2
)
(
2
2
)
CDF of different fade distributions are listed in [9, Table 3, pp. 420]. Therefore, for Nakagami-m and Rice i.n.d channel, (26) becomes, SC − Nak
= 1−
∫
λ λ e− (γ + ) I u 2γ u
2
2
0
(
2γλ
)∏ L
l =1
Γ( ml , Ωm γ ) 1 − d γ (27) Γ(ml )
SC − Ric
∫
Pd
= 1−
∫
∞
x
∏ l =1
u
2
2
0
L
λ − ( γ + λ ) e Iu 2γ
SC − Nak
2γλ
) ,
2(1 + K l ) 1 − Q1 2 K l , γ Ωl
(28)
(
∞
Pd
SC
Pd
SC − Nak
∫
×
Pd
∞
)
L −1
dγ
γ
0
L G (u + k , λ2 ) ∞ k −γ = 1− ∑ ∫0 γ e fγ (γ ) Fγ (γ ) k = 0 k ! Γ (u + k )
(
)
L −1
dγ
using
SC − Nak
∫
m )γ Ω
− (1+
[10,
L −1
∑
m )γ Ω
q =1
(−1) ( e q! q
Pd
SC − Nak
L m Γ(m) Ω
= 1−
m ∞
− q Ωm γ
m −1 m j ) (Ωγ) ∑ j =0 j !
produces
q
dγ
k =0
q ( m −1)
×
∑ζ
j
1 G (u + k , λ2 ) L −1 (−1) q Γ(u + k ) q =1 q !
∑ k!
(35)
(m, q, Ω)
j =0
∫
∞
0
γ
∑
m + k + j −1 − (1+ Ωm (1+ q )) γ
e
(36)
dγ
where ζ j (m, q, Ω) is the coefficient of γ j in the multinomial expansion of
m −1 ( m γ ) j ∑ Ω j =0 j !
m Ω
j
q
which is given by
βi ( q −1)
j
∑
ζ j (m, q, Ω) =
I[0,( q −1)( m −1) (q), ( j − i)!
i = j − m +1
(37)
a≤b≤c
1 I[ a ,c ] (b) = 0
otherwise
where β00 = β0 q = 1,
β j1 = 1 j !,
β1q = q .
Using the identity in [10, 3.326-2], (36) then yields, Pd
SC − Nak
= 1−
L m Γ(m) Ω
m ∞
∑ζ
k =0
(m, q, Ω)
j
3.2.2
1 G (u + k , λ2 ) L −1 (−1)q ∑ Γ(u + k ) q =1 q !
∑ k!
q ( m −1)
(31)
8.352-4]
1 G (u + k , λ2 ) ∑ k = 0 k ! Γ (u + k )
j =0
L −1
γ m + k −1e − (1+
identity ∞
which after multinomial expansion becomes
(30)
Γ(m, Ωm γ ) 1 − dγ 0 Γ ( m) For the special case of dual diversity combiner, (31) becomes ∞
∑
the m
(34) q
(−1) q Γ( m, Ωm γ ) dγ q ! Γ(m) q =1
L −1
0
SC − Nak
(33)
1 G (u + k , λ2 ) Γ(u + k )
k =0
L m = 1− Γ(m) Ω
×
For Nakagami-m channel, (33) becomes m L m ∞ 1 G (u + k , λ2 ) Pd = 1− ∑ Γ(m) Ω k = 0 k ! Γ(u + k )
1 Γ(u + k , λ2 ) ∑ k = 0 k ! Γ (u + k )
∑ k!
e
× ∫ γ m + k −1e
(29)
(32)
m+2 ∞
m ∞
m + k −1 − (1+ Ωm )γ
Case 1: Nakagami-m fading channel
×
2
m Ω
L m Γ(m) Ω
= 1−
∞
Going forward, we split the derivation for Nakagami and Rice fading channel; though the same principle is applicable to both in order to derive more compact solution for some special cases.
3.2.1
k =0
− (1+ Ωm ) γ
When m is an integer, (31) can be expressed using Binomial expansion as
which after substituting (8) becomes ∞
m + k −1
( Γ ( m) )
probability using (23) is given by Pd = ∫ LQu ( 2γ , λ ) fγ (γ ) Fγ (γ )
1 G (u + k , λ2 ) Γ(u + k )
∞
∑ k!
where 2 F1 (.,.;.;.) is Gauss hypergeometric function.
d γ
SC
0
∞
l
Eq. (26) can be solved numerically using numerical integration methods with the aid of computer (i.e. using MATLAB or Mathematica) and it is easier to program, more compact and general across fading channels than the result in [5, (17), (21)] which holds for only Nakagami-m channel with dual SDC combiner and multiple combiner with integer fading index, m . For the i.i.d channel scenario, the PDF is given by L −1 fγ (γ ) = Lfγ (γ ) ( Fγ (γ ) ) and therefore, the average detection
SC
m
Γ(2m + k ) m F 1, 2m + k ; m + 1; × 2m 2 m+ k 2 1 2m + Ω m(1 + Ω )
which
(
2 m Γ(m) Ω
2
= 1−
l
and Pd
= 1−
G (m, Ωm γ ) × γ e dγ 0 Γ ( m) Using the identity in [10, 6.455-2], we obtain
l
SC
Pd
SC − Nak
l =1
Eq. (26) can be evaluated numerically to obtain Pd . The
∞
Pd
(38)
Γ( m + k + j )
(1 + Ωm (1 + q) )
m+k + j
Case 2: Rice fading channel
For Rice channel, (30) becomes Pd
SC − Ric
= 1−
∞
L G (u + r , λ2 ) Γ(u + r )
∑ r! k =0
∫
∞
0
γ r e−γ
1+ K (1 + K )γ exp − K − Ω Ω
2(1 + K ) × I o 2 K (1 + K )γ / Ω 1 − Q1 2 K , γ Ω
(
)
(39)
L −1
dγ
For special case of dual selection combiner (38) becomes Pd SC −Ric = 1 −
∫
∑ r =0
2 G (u + r , λ2 ) 1 + K −K ∂r (−1)φγ( r ) ( s) |s =1 − e (−1) r r r ! Γ(u + r ) Ω ∂s
(1+ K )γ ∞ − s + Ω
0
e
2(1 + K ) I o 2 K (1 + K )γ / Ω Q1 2 K , γ Ω
(
)
(40) d γ s =1
which after manipulation and using [13, eq. (46)], we obtain ∞ 2 G (u + r , λ2 ) Pd = 1− ∑ (−1)φγ( r ) ( s ) |s =1 − 2 Ae− K r ! Γ ( u + r ) r =0 (41) r r ∂ × (−1) r I 2 A , 2 K , 2 AK , 2( s + A ∂s s =1 1+ K where A = and Ω 2 1 c d ac − I (a, b, c, d ) = e 2 d Q1 b , 2 2 (d + a ) d ( d + a ) d .
(
SC − Ric
)
(
)
c2 −b2d
×
abc 2 a2 e 2( d +a ) I 0 2 d (d + a 2 ) (d + a )
Substituting φγ( r ) ( s ) |s =1 for Rice fading we obtain, ∞
PdSC−Ric = 1 −
2 G (u + r , λ2 ) Ωr (r !)(1 + K ) 1 −K Ω r exp 2 Γ(u + r ) (1 + K + Ω)1+ r 1 + K + Ω i = 0 (i !) (r − 1)! (42)
∑ r! r =0
i
K (1 + K ) r −K × − (−1) 2 Ae 1+ K + Ω
4 and it confirms the observation in Fig. 3. Also, Fig. 4 indicates that lower number of samples favours the SLS diversity scheme more than the SDC. This is expected as the performance of SLS is more affected by the sampled signal, because the selection of best detection path or circuitry is done after sampling unlike the SDC, in which the best path is selected before sampling. However, in practical implementation, SLS requires more number of detection circuitry than SDC which requires only one detection circuitry.
∑
0
10
Probability of Miss Detection Pm
∞
-1
10
-2
10
L=2, L=2, L=2, L=2, L=4, L=4, L=4, L=4,
-3
10
∂ r I 2 A , 2 K , 2 AK , 2(s + A ∂ s s =1 r
(
)
The rth derivative of I (a, b, c, d ) could be computed using MAPLE. For higher order configuration, the derivation becomes involved and therefore it is recommended to use numerical evaluation in (28) derived for i.n.d Rice channel.
In this article, we analyse the performance of energy detector using the complementary ROC curves and detection probability. Here, we focus on the Rice fading as there is limited information on the performance of energy detector in Rice fading in literature. First we study the effect of detection parameters on the detection performance of SLS and SDC diversity schemes and then compare the two selection schemes. Fig. 1 and 2 shows the effect of change in number of diversity branches, L, fading index, K, and time-bandwidth product, u (which is a function of number of samples). It is observed that that both diversity schemes perform better as the number of diversity branches and fading index is increases. Also, lower number of samples improves the performance. The effect of change in different combinations of the three parameters can also be observed from both figures. Fig. 3 shows the comparison of the two diversity schemes for different mean SNR values. We observe that the choice of better diversity scheme for energy detection varies with mean SNR. At low mean SNR, SDC is the better scheme but as the mean SNR increases, the performance of SLS becomes better than that of SDC while high mean SNR values favours again the SDC scheme. Due to this overlap, we decided to investigate the variation of average detection probability, Pd with mean SNR for both schemes. The result is shown in Fig.
-4
10
u=0.5 u=5.5 u=0.5 u=5.5 u=0.5 u=5.5 u=0.5 u=5.5
-3
-2
-1
10 10 10 Probability of False Alarm P f
0
10
Fig. 1 Complementary ROC curves of SLS diversity scheme over Rice channels K= {1, 4}, for u= {0, 5.5}, L ={2, 4}, and mean SNR=10 dB.
0
10
Probability of Miss Detection Pm
4 Numerical results
-4
10
K=1, K=1, K=4, K=4, K=1, K=1, K=4, K=4,
-1
10
-2
10
L=2, L=2, L=2, L=2, L=4, L=4, L=4, L=4,
-3
10
K=1, K=1, K=4, K=4, K=1, K=1, K=4, K=4,
u=0.5 u=5.5 u=0.5 u=5.5 u=0.5 u=5.5 u=0.5 u=5.5
-4
10
-4
10
-3
-2
-1
10 10 10 Probability of False Alarm Pf
0
10
Fig. 2 Complementary ROC curves of SDC diversity scheme over Rice channels K= {1, 4}, for u= {0, 5.5}, L = {2, 4}, and mean SNR=10 dB.
scheme (for branches with i.n.d statistics) with closed form expressions derived for special cases. With these mathematical frameworks, we are able to show that the choice of better detection diversity scheme between SLS and SDC depends on the mean SNR values. Careful consideration of related publications revealed that this simplified approaches have not been applied in analyzing the performance of the energy detector until now and to the best of our knowledge, limited analysis has been carried out on the performance of energy detector in Rice fading channel. We have shown that our approach, unlike other methods which are incapable of handling half-integer u and Lu, produces result for these cases. These results could be readily used in deciding suitable diversity scheme, number of diversity branches and the energy threshold value required to achieve a specified false alarm and detection rate for different scenario of energy detector receiver in cognitive radio system.
0
Probability of Miss Detection Pm
10
-1
10
-2
10
SNR = SNR = SNR = SNR = SDC SLS
-3
10
-4
10
-4
10
-3
0dB 5dB 10dB 15dB
-2
-1
10 10 10 Probability of False Alarm Pf
0
10
Fig. 3 Complementary ROC curves for dual SLS and SDC over Rice channels (K=2) for u=2.5 and mean SNR= {0 ,5, 10, 15}.
Probability of Detection Pd
1
Acknowledgment This work is supported in part by funding from the US Army Research Office (W911NF-04-2-0054), Clarkson Aerospace, and National Science Foundation (0931679 & 1040207).
References
0.9
[1]
0.8
[2]
0.7
[3]
0.6
[4] 0.5 L=2, u=0.5 L=2, u=2.5 L=2, u=5.5 L=4, u=0.5 L=4, u=2.5 L=4, u=5.5 SDC SLS
0.4 0.3 0.2 0.1 0 0
5
10 Mean SNR (dB)
15
[5]
[6]
[7] 20
Fig.4 Average probability of detection for SLS and SDC over Rice channel (K=2) for L= {2,4}, u= {0.5, 5}, and Pf=0.01.
[8]
[9] [10]
5 Conclusions In this paper, the detection an unknown deterministic signal by using an energy detector has been considered with detectors’ selection diversity system. Both SDC and SLS diversity have been analyzed. We derived an expression for the performance analysis of both diversity systems. The derivation for SLS performance is based on the canonical series form of Marcum Q-function in conjunction with derivatives of MGF of SNR while a novel technique resulting into a single integral expression was presented for SDC
[11]
[12]
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