On the Energy Detection of Unknown Deterministic ...

On the Energy Detection of Unknown Deterministic Signal over Nakagami Channels with Selection Combining Sanjeewa Herath, Nandana Rajatheva Asian Institute of Technology, Thailand.

Chintha Tellambura University of Alberta, Canada.

CCECE 05th May 2009

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Presentation Outline Motivation System Model Detection Problem Fading Diversity Novel Derivations Fading - Nakagami-m Selection Combiner - Nakagami-m Numerical Results Conclusion

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Why Spectrum Sensing?

Figure: Spectrum Hole (a) Space (b) Time [1] ◮

Secondary (unlicensed) users are allowed to utilize spectrum holes opportunistically ◮ ◮

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Improves spectrum efficiency Spectrum scarcity is resolved to certain extend

Spectrum sensing with Energy detection

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Why Energy Detection? ◮

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Primary transmission is modeled as an unknown deterministic signal Privacy of primary transmissions is guaranteed ◮

◮ ◮ ◮ ◮

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Detect just existence of transmissions in the spectrum band of interest Less priori knowledge (Signal shape, duration etc not used) Inherent security to primary communication - not tapped Simple in structure - we will see Detect any shape of signal

Detect the received energy over a period and decide (sub optimal) Performance variation over different wireless environment is important ◮ ◮

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Quantify the performance reduction over Fading Overcome with receiver diversity, user cooperation

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Detector Model

Figure: Energy Detection

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Detection is a test of binary hypothesis ([2]) ◮ ◮

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H0 : Primary transmission does not exist (noise) - χ22u H1 : Primary transmission exist - χ22u (ǫ)

Selection diversity - Select the branch with highest SNR ◮ ◮

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Simple structure is preserved - minor modification Performance is improved

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Detection ◮

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Detection probability (Pd ) (Conditional) [3] p √  P d = Qu 2γ, λ False alarm probability (Pf )

Γ u, λ2 Pf = Γ(u) ◮

(1)




Average detection probability (P d ) Z ∞ p √ Pd = Qu ( 2γ, λ)fγ (γ) d γ

(2)

(3)

0

◮

False alarm - Does not depend on SNR ⇒ fading

u - Time bandwidth product (integers - by choice) Qu (., .) - u th order Marcum Q-function γ - SNR; λ - Energy threshold Γ(., .) - incomplete gamma function CCECE 2009 : AIT

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Detection over Fading Channels ◮

Existing results over fading channels ◮ ◮ ◮

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Integrals found involving Marcum-Q ◮ ◮ ◮

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Rayleigh fading - [3, 4, 5] Rician fading - [3, 5]-u = 1 Nakagami-m fading [3]-integer m, [5]-integral Product of Marcum-Q, Bessel, exponential and rational of γ Uses limited amount of results available [6, 7] Difficult to evaluate in general

In this paper: Detection over Nakagami-m fading alternative result ◮ ◮ ◮

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Use of alternative representations of Marcum-Q function Introduced in [8] for analyzing EGC (our result) Transform the integrals to other forms - relatively tractable

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Detection over Diversity Receivers ◮

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Find the conditional detection probability and average over the respective PDF of output SNR Available results over Rayleigh branches ◮ ◮ ◮ ◮ ◮

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Selection Combining (SC) [3], [4] Maximal Ratio Combining (MRC) [3], [4] Equal gain diversity combining (EGC) [8] - Nakagami-m Switch and Stay Combining (SSC) [3] Square-Law Combining Schemes(SLC, SLS) [3]

All diversity results are limited to Rayleigh fading - except our results in [8] In this paper: Selection Combining over Nakagami-m fading ◮ ◮

Less complex in implementation Quantify the performance gain/loss over ◮ ◮

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Number of branches L Other parameters m, u, γ

Difficulty of integrals? overcome by transforms using alternate representations

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Detection over Nakagami-m Fading ◮

Alternate representation [9] ∞ X p √ γ n e−γ Qu ( 2γ, λ) = n! n=0

n+u−1 X k =0

λ

e− 2 k!

 k λ 2

(4)

Avg. detection probability P d,Nak m X n  k λ  ∞  n+u−1 m γ (n + m − 1)! X 1 λ e− 2 P d,Nak = Γ(m) γ + m γ+m n! k! 2 ◮

k =0

n=0

Error bound | ENak | #  m "   X n N  m γ γ (m)n | ENak |≤ − 1 F0 m; ; γ+m γ+m γ+m n!

◮

n=0

Hypergeometric series - p Fq (a1 , ..., ap ; b1 , ..., bq ; x) Pochhammer symbol - (a)n

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Detection over Selection Combining ◮

Alternate representation [9] n ∞  p X √  λ 2 p −(γ+ λ2 ) Qu 2γ, λ = 1 − e In ( 2λγ) 2γ n=u

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Avg. Detection over dual branch any m P d,sc,2

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(5)

 2m X  n λ ∞ 2e− 2 Γ(2m) m 1 λ =1 − m Γ2 (m) γ + 2m n! 2 n=u   m λγ × Ψ1 2m, 1; m + 1, n + 1; , γ + 2m 2(γ + 2m)

(6)

Error bound | Esc,2 | | Esc,2

 2m  n ! λ N X λ 2 e− 2 Γ(2m) m 1 λ |≤ e2 − m Γ2 (m) γ + 2m n! 2 n=0   m λγ × Ψ1 2m, 1; m + 1, N + 1; , γ + 2m 2(γ + 2m)

(7)

Horns function - Ψ1 (α, β; γ, γ´ ; x, y) CCECE 2009 : AIT

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Detection over Selection Combining cont... ◮

Avg. Detection over integer m any number of branches λ

P d,sc,L =1 − Le− 2



m γ

k (m−1)

k =0

X ζi (m, k , γ) (m)i

× ◮

m X   n ∞ X L−1  L−1 λ k 1 (−1) k n! 2 n=u (i+m)

βL

i=0

Error bound | Esc,L | | Esc,L |≤L e

− λ2



m γ

k (m−1)

m "

(8)

  λ 1 F1 i + m; n + 1; 2βL

 n # X  N L−1  X 1 λ L−1 e − n! 2 k λ 2

n=0

k =0

  X ζi (m, k , γ) (m)i λ × 1 F1 i + m; N + 1; (i+m) 2βL βL i=0

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False Alarm Probability ◮

Pick the branch with maximum SNR - max(γ1 , γ2 , ..., γL )

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SNR in each branch - γl =

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Select the branch with maximum hl

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Under H0 : Samples from noise (does not matter branch selection)

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The statistics of decision variable is χ22u

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False alarm probability (tail of χ22u )   Γ u, λ2 P f ,sc,L = Γ(u)

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hl2 Es N01 , l

= 1, 2, ..., L

(9)

Nothing to average

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Convergence of Series ◮

Truncation error 0

10

−2

10

−4

10

Error Bound

−6

10

−8

10

−10

10

|ENak|

−12

10

|E

|

sc,2

−14

10

|Esc,L|, L = 2 |E

|, L = 3

sc,L

−16

10

0

10

1

2

10

10

3

10

Number of terms − N

Figure: Error Bounds u = 1, m = 2, SNR = 10 dB, L = 2 and Pf = 0.01 CCECE 2009 : AIT

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Detector Performance - SNR 0

10

−1

Probability of a Miss Pm

10

−2

10

−3

10

−4

10

SNR = 0dB SNR = 10dB SNR = 15dB SNR = 20dB Simulation

−5

10

−6

10

−4

10

−3

10

−2

−1

10 10 Probability of a False Alarm P

0

10

f

Figure: Complementary ROC curves over Nakagami-m fading channel (u = 1, m = 2, SNR = {0, 10, 15, 20} dB)

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Detector Performance - m 0

10

−1

Probability of a Miss Pm

10

−2

10

−3

10

m=1 m=2 m=3 m=4 Simulation

−4

10

−5

10

−4

10

−3

10

−2

−1

10 10 Probability of a False Alarm P

0

10

f

Figure: Complementary ROC curves over Nakagami-m fading channel (u = 1, m = {1, 2, 3, 4}, SNR = 10 dB)

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Detector Performance - Branches 0

10

−1

Probability of a Miss Pm

10

−2

10

−3

10

−4

10

L=1 L=2 L=3 L=4 Simulation

−5

10

−6

10

−4

10

−3

10

−2

−1

10 10 Probability of a False Alarm P

0

10

f

Figure: Complementary ROC curves of SC receiver over Nakagami-m fading channel (L = {1, 2, 3, 4}, u = 2, m = 2, SNR = 10 dB) CCECE 2009 : AIT

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Conclusion ◮

Energy Detection over Fading ◮ ◮

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Energy Detection over SC ◮ ◮

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Nakagami-m Fading Exact detection probability

Use of alternative representation of Marcum-Q function for evaluation of related integral Simulation results ◮

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Nakagami-m Fading Exact detection probability, Alternative expression

support the decision variable formulation, derivations

Helpful in designing and evaluating ◮ ◮

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Cognitive radio, Ultra wide-band Detector threshold for expected false alarm rate and different u and γ

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Thank You 5th May 2009

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References I T. Rahul, M. Mubaraq, and S. Anant, “Extended edition: What is a spectrum hole and what does it take to recognize one?” UC Berkeley EECS, Tech. Rep. UCB/EECS-2008-110, Aug. 2008. H. Urkowitz, “Energy detection of unknown deterministic signals,” Proc. IEEE, vol. 55, no. 4, pp. 523–531, 1967. F. F. Digham, M.-S. Alouini, and M. K. Simon, “On the energy detection of unknown signals over fading channels,” IEEE Trans. Commun., vol. 55, no. 1, pp. 21–24, Jan. 2007. A. Pandharipande and J.-P. Linnartz, “Performance analysis of primary user detection in a multiple antenna cognitive radio,” Proc. IEEE Int. Conf. Communications (ICC), pp. 6482–6486, 2007. V. Kostylev, “Energy detection of a signal with random amplitude,” Proc. IEEE Int. Conf. Communications (ICC), vol. 3, pp. 1606–1610, 2002. A. H. Nuttall, “Some integrals involving the Q function,” in Naval Underwater Systems Center (NUSC) technical report, Apr. 1972.

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References II ——, “Some integrals involving the QM function,” in Naval Underwater Systems Center (NUSC) technical report, May 1974. S. P. Herath and N. Rajatheva, “Analysis of equal gain combining in energy detection for cognitive radio over Nakagami channels,” IEEE Global Telecommn. Conf. (GLOBECOM), pp. 1–5, 2008. M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels, 2nd ed. New York: Wiley, 2005.

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