Spectrum Sensing in Cognitive Radios using Distributed Sequential Detection

Transcription

1 Spectrum Sensing in Cognitive Radios using Distributed Sequential Detection Jithin K. S. Research Supervisor: Prof. Vinod Sharma External Examiner: Prof. Shankar Prakriya, IIT Delhi ECE Dept., Indian Institute of Science, Bangalore February 4, 2013

2 Outline Introduction Parametric Distributed Sequential Tests Nonparametric Distributed Sequential Tests Multihypothesis scenario Conclusions

3 Introduction

4 Introduction CR: Radio systems, which perform Radio Environment Analysis, identify spectral holes and operate in those holes.

5 Introduction CR: Radio systems, which perform Radio Environment Analysis, identify spectral holes and operate in those holes. Licensed (Primary) and Unlicensed (Secondary) users

6 Introduction CR: Radio systems, which perform Radio Environment Analysis, identify spectral holes and operate in those holes. Licensed (Primary) and Unlicensed (Secondary) users Spectral holes in various dimensions

7 Introduction CR: Radio systems, which perform Radio Environment Analysis, identify spectral holes and operate in those holes. Licensed (Primary) and Unlicensed (Secondary) users Spectral holes in various dimensions Aided by evolving SDR technology

8 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission

9 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present

10 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present Primary Techniques:

11 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present Primary Techniques: Matched filter

12 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present Primary Techniques: Matched filter Cyclostationary detector

13 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present Primary Techniques: Matched filter Cyclostationary detector Energy detector

14 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present Primary Techniques: Matched filter Cyclostationary detector Energy detector Challenges:

15 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present Primary Techniques: Matched filter Cyclostationary detector Energy detector Challenges: Shadowing, Fading and low SNR

16 Spectrum Sensing Identify spectral holes and quickly detect the onset of primary transmission Spectrum Sensing Hypothesis testing formulation: H 0 : Primary user transmission is absent H 1 : Primary user transmission is present Primary Techniques: Matched filter Cyclostationary detector Energy detector Challenges: Shadowing, Fading and low SNR Sensing Frequency and duration

17 Cooperative setup Mitigates multipath fading, shadowing and hidden node problem. Improves Probability of Errors (P FA and P MD ) and Expected Detection Delay (E DD ) A distributed statistical inference problem

18 Cooperative setup Mitigates multipath fading, shadowing and hidden node problem. Improves Probability of Errors (P FA and P MD ) and Expected Detection Delay (E DD ) A distributed statistical inference problem dbm PU Transmit Power Sensitivity Level Loss due to Path Loss Threshold with Cooperation Loss due to Multipath fading and Shadowing Improvement with Cooperation Noise Floor Threshold without Cooperation

19 Problems addressed in this thesis Distributed Detection:

20 Problems addressed in this thesis Distributed Detection:

21 Problems addressed in this thesis Distributed Detection:

22 Problems addressed in this thesis Distributed Detection:

23 Problems addressed in this thesis Distributed Detection: centralized

24 Problems addressed in this thesis Distributed Detection: centralized/decentralized

25 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR:

26 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous

27 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous

28 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous

29 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous

30 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous /Asynchronous

31 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous /Asynchronous

32 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous /Asynchronous

33 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous /Asynchronous

34 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous /Asynchronous Decision Mechanism: fixed sample size

35 Problems addressed in this thesis Distributed Detection: centralized/decentralized Sampling at CR: synchronous /Asynchronous Decision Mechanism: fixed sample size /Sequential

36 Problems addressed in this thesis (Contd.) Focus on reducing the number of samples in the cooperative setup with reliability constraints (Decentralized Sequential Hypothesis Testing)

37 Problems addressed in this thesis (Contd.) Focus on reducing the number of samples in the cooperative setup with reliability constraints (Decentralized Sequential Hypothesis Testing) Fast detection of idle TV bands

38 Problems addressed in this thesis (Contd.) Focus on reducing the number of samples in the cooperative setup with reliability constraints (Decentralized Sequential Hypothesis Testing) Fast detection of idle TV bands Quickest detection of unoccupied spectrum in slotted primary user transmission systems

39 Problems addressed in this thesis (Contd.) Focus on reducing the number of samples in the cooperative setup with reliability constraints (Decentralized Sequential Hypothesis Testing) Fast detection of idle TV bands Quickest detection of unoccupied spectrum in slotted primary user transmission systems Imprecise estimates of the channel gains and noise power (parameter uncertainty-composite Sequential Testing)

40 Problems addressed in this thesis (Contd.) Focus on reducing the number of samples in the cooperative setup with reliability constraints (Decentralized Sequential Hypothesis Testing) Fast detection of idle TV bands Quickest detection of unoccupied spectrum in slotted primary user transmission systems Imprecise estimates of the channel gains and noise power (parameter uncertainty-composite Sequential Testing) Channel gain statistic unknown (Universal Sequential Testing)

41 Problems addressed in this thesis (Contd.) Focus on reducing the number of samples in the cooperative setup with reliability constraints (Decentralized Sequential Hypothesis Testing) Fast detection of idle TV bands Quickest detection of unoccupied spectrum in slotted primary user transmission systems Imprecise estimates of the channel gains and noise power (parameter uncertainty-composite Sequential Testing) Channel gain statistic unknown (Universal Sequential Testing) Detect the spectrum and identify the primary user (Decentralized Multihypothesis Sequential Testing)

42 Related Work Sequential tests: Wald 47, Irle 84 Decentralized Sequential tests: Mei 08, Fellouris et al. 12, Yilmaz et al. 12 Cooperative Spectrum Sensing and Censored detection: Akyildiz et al. 11 Universal hypothesis tests: Levitan et al. 02, Jacob et al. 08, Unnikrishnan et al. 11 Multihypothesis Sequential tests: Draglia et al. 99, Tartakovsky 00

43 Model N k,1 h 1 + X k,1 Processing Y k,1 N k,2 h 2 Primary User X k,2 Processing Y k,2 S k h L +. N k,l + X k,l. Processing Y k,l Secondary Fusion Center

44 Model N k,1 h 1 + X k,1 Processing Y k,1 N k,2 h 2 Primary User X k,2 Processing Y k,2 S k h L +. N k,l + X k,l. Processing Y k,l Secondary Fusion Center General Problem: N = Stopping time, δ N = decision rule min E[N H 0 ] and min E[N H 1 ], N,δ N N,δ N subject to P FA α 1 and P MD α 2

45 Parametric Distributed Sequential Tests At l th CR, H 0 : X k,l f 0,l H 1 : X k,l f 1,l f 0,l and f 1,l are fully known f 0,l and f 1,l are not fully known: parametric family.

46 DualSPRT Algorithm 1. At Node, l, Sequential Probability Ratio Test (SPRT), W k,l = W k 1,l + log f 1,l(X k,l ) f 0,l (X k,l ), W 0,l = 0.

47 DualSPRT Algorithm 1. At Node, l, Sequential Probability Ratio Test (SPRT), 2. Node l transmits W k,l = W k 1,l + log f 1,l(X k,l ) f 0,l (X k,l ), W 0,l = 0. Y k,l = b 1 1 {Wk,l γ 1} + b 0 1 {Wk,l γ 0}

48 DualSPRT Algorithm 1. At Node, l, Sequential Probability Ratio Test (SPRT), 2. Node l transmits W k,l = W k 1,l + log f 1,l(X k,l ) f 0,l (X k,l ), W 0,l = 0. Y k,l = b 1 1 {Wk,l γ 1} + b 0 1 {Wk,l γ 0} Noisy MAC at the FC, Z k i.i.d. MAC noise Y k = L Y k,l + Z k l=1

49 DualSPRT Algorithm 1. At Node, l, Sequential Probability Ratio Test (SPRT), 2. Node l transmits W k,l = W k 1,l + log f 1,l(X k,l ) f 0,l (X k,l ), W 0,l = 0. Y k,l = b 1 1 {Wk,l γ 1} + b 0 1 {Wk,l γ 0} Noisy MAC at the FC, Z k i.i.d. MAC noise Y k = L Y k,l + Z k l=1 3. FC runs SPRT: g µ is p.d.f. of µ + Z k

50 DualSPRT Algorithm 1. At Node, l, Sequential Probability Ratio Test (SPRT), 2. Node l transmits W k,l = W k 1,l + log f 1,l(X k,l ) f 0,l (X k,l ), W 0,l = 0. Y k,l = b 1 1 {Wk,l γ 1} + b 0 1 {Wk,l γ 0} Noisy MAC at the FC, Z k i.i.d. MAC noise Y k = L Y k,l + Z k l=1 3. FC runs SPRT: g µ is p.d.f. of µ + Z k F k = F k 1 + log g µ 1 (Y k ) g µ0 (Y k ), F 0 = 0,

51 DualSPRT Algorithm 1. At Node, l, Sequential Probability Ratio Test (SPRT), 2. Node l transmits W k,l = W k 1,l + log f 1,l(X k,l ) f 0,l (X k,l ), W 0,l = 0. Y k,l = b 1 1 {Wk,l γ 1} + b 0 1 {Wk,l γ 0} Noisy MAC at the FC, Z k i.i.d. MAC noise Y k = L Y k,l + Z k l=1 3. FC runs SPRT: g µ is p.d.f. of µ + Z k F k = F k 1 + log g µ 1 (Y k ) g µ0 (Y k ), F 0 = 0, 4. FC decides, at N = inf{k : F k / ( β 0, β 1 )}, H 1 if F N β 1, H 0 if F N β 0.

52 Performance Analysis of DualSPRT- E DD Sample path argument W k,l γ LLR Sum at local node l under H 1 γ T ime Transmission from local node l Mean of the received signal at FC F k β β LLR Sum at FC under H 1

53 Performance Analysis of DualSPRT- E DD Sample path argument W k,l γ LLR Sum at local node l under H 1 γ b N 1 l T ime Transmission from local node l b Mean of the received signal at FC β (1 node transmits) t 1 F k β LLR Sum at FC under H 1

54 Performance Analysis of DualSPRT- E DD Sample path argument W k,l γ LLR Sum at local node l under H 1 γ b N 1 l T ime Transmission from local node l b Mean of the received signal at FC β (1 node transmits) t 1 t 2 F k β LLR Sum at FC under H 1

55 Performance Analysis of DualSPRT- E DD Sample path argument W k,l γ lim γ P1(Nl = N l 1 ) 1, lim Nl/γ = lim N l 1 /γ γ γ lim γ,β P1(N = N 1 ) 1, lim γ,β N/γ = lim γ,β N 1 /γ LLR Sum at local node l under H 1 γ b N 1 l T ime Transmission from local node l b Mean of the received signal at FC β (1 node transmits) t 1 t 2 t 3 t l F k β P1(Nl = Nl i ) 1, Nl N 1 LLR Sum at FC under H 1

56 Performance Analysis of DualSPRT- E DD (Contd.) At each node l, time to cross the threshold N 1 l N ( γ, ρ2 l γ δ l δl 3 ), δ l = E 1 [LLR l ] and ρ 2 l = Var 1 [LLR l ]

57 Performance Analysis of DualSPRT- E DD (Contd.) At each node l, time to cross the threshold N 1 l N ( γ, ρ2 l γ δ l δl 3 ), δ l = E 1 [LLR l ] and ρ 2 l = Var 1 [LLR l ] t j = jth order statistics of N 1 1, N 1 2,..., N 1 L

58 Performance Analysis of DualSPRT- E DD (Contd.) At each node l, time to cross the threshold N 1 l N ( γ, ρ2 l γ δ l δl 3 ), δ l = E 1 [LLR l ] and ρ 2 l = Var 1 [LLR l ] t j = jth order statistics of N 1 1, N 1 2,..., N 1 L l = min{j : δ j FC > 0 and β F j δ j FC < E[t j+1 ] E[t j ]}

59 Performance Analysis of DualSPRT- E DD (Contd.) At each node l, time to cross the threshold N 1 l N ( γ, ρ2 l γ δ l δl 3 ), δ l = E 1 [LLR l ] and ρ 2 l = Var 1 [LLR l ] t j = jth order statistics of N 1 1, N 1 2,..., N 1 L l = min{j : δ j FC > 0 and β F j δ j FC < E[t j+1 ] E[t j ]} Fj = E[F tj 1], F j = F j 1 + δ j FC (E[t j] E[t j 1 ]), F 0 = 0. E DD (E 1 [N]) E 1 [N 1 ] E[t l ] + β F l δ l FC

60 Performance Analysis of DualSPRT- P MD P MD = P 1 (N 0 < N 1 ) P 1 (N 0 < t 1, N 1 > t 1 )

61 Performance Analysis of DualSPRT- P MD P MD = P 1 (N 0 < N 1 ) P 1 (N 0 < t 1, N 1 > t 1 ) P 1 (N 0 < t 1 )

62 Performance Analysis of DualSPRT- P MD P MD = P 1 (N 0 < N 1 ) P 1 (N 0 < t 1, N 1 > t 1 ) P 1 (N 0 < t 1 ) Also P 1 (N 0 < N 1 ) P 1 (N 0 < )

63 Performance Analysis of DualSPRT- P MD P MD = P 1 (N 0 < N 1 ) P 1 (N 0 < t 1, N 1 > t 1 ) P 1 (N 0 < t 1 ) Also P 1 (N 0 < N 1 ) P 1 (N 0 < ) = P 1 (N 0 < t 1 ) + P 1 (t 1 N 0 < t 2 ) + P 1 (t 2 N 0 < t 3 ) +...

64 Performance Analysis of DualSPRT- P MD P MD = P 1 (N 0 < N 1 ) P 1 (N 0 < t 1, N 1 > t 1 ) P 1 (N 0 < t 1 ) Also P 1 (N 0 < N 1 ) P 1 (N 0 < ) = P 1 (N 0 < t 1 ) + P 1 (t 1 N 0 < t 2 ) + P 1 (t 2 N 0 < t 3 ) +... P MD P 1 (N 0 < t 1 )

65 Performance Analysis of DualSPRT- P MD P MD = P 1 (N 0 < N 1 ) P 1 (N 0 < t 1, N 1 > t 1 ) P 1 (N 0 < t 1 ) Also P 1 (N 0 < N 1 ) P 1 (N 0 < ) = P 1 (N 0 < t 1 ) + P 1 (t 1 N 0 < t 2 ) + P 1 (t 2 N 0 < t 3 ) +... P MD P 1 (N 0 < t 1 ) = P k=1 [ {F k < β} k 1 n=1 {F n > θ} t 1 > k ] P[t 1 > k]

66 Comparison of Analysis with Simulations 5 Secondary nodes channel gains (0, -1.5, -2.5, -4 and -6 db)

67 Comparison of Analysis with Simulations 5 Secondary nodes channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1

68 Comparison of Analysis with Simulations 5 Secondary nodes channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1 P MD Sim. P MD Anal. E DD Sim. E DD Anal e e e e e e

69 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}]

70 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}] L L Dtot 0 = D(f 0,l f 1,l ), Dtot 1 = D(f 1,l f 0,l ), r l = D(f 0,l f 1,l ), ρ Dtot 0 l = D(f 1,l f 0,l ) Dtot 1 l=1 l=1

71 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}] L L Dtot 0 = D(f 0,l f 1,l ), Dtot 1 = D(f 1,l f 0,l ), r l = D(f 0,l f 1,l ), ρ Dtot 0 l = D(f 1,l f 0,l ) Dtot 1 l=1 Theorem l=1 γ 0,l = r l log c,γ 1,l = ρ l log c, β 0 = log c, β 1 = log c.

72 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}] L L Dtot 0 = D(f 0,l f 1,l ), Dtot 1 = D(f 1,l f 0,l ), r l = D(f 0,l f 1,l ), ρ Dtot 0 l = D(f 1,l f 0,l ) Dtot 1 l=1 Theorem l=1 γ 0,l = r l log c,γ 1,l = ρ l log c, β 0 = log c, β 1 = log c. E 0 [N] lim c 0 log c 1 E 1 [N] Dtot 0 + M 0 and lim c 0 log c 1 Dtot 1 + M 1

73 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}] L L Dtot 0 = D(f 0,l f 1,l ), Dtot 1 = D(f 1,l f 0,l ), r l = D(f 0,l f 1,l ), ρ Dtot 0 l = D(f 1,l f 0,l ) Dtot 1 l=1 Theorem l=1 γ 0,l = r l log c,γ 1,l = ρ l log c, β 0 = log c, β 1 = log c. E 0 [N] lim c 0 log c 1 E 1 [N] Dtot 0 + M 0 and lim c 0 log c 1 Dtot 1 + M 1 Gaussian MAC noise at FC µ chosen appropriately

74 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}] L L Dtot 0 = D(f 0,l f 1,l ), Dtot 1 = D(f 1,l f 0,l ), r l = D(f 0,l f 1,l ), ρ Dtot 0 l = D(f 1,l f 0,l ) Dtot 1 l=1 Theorem l=1 γ 0,l = r l log c,γ 1,l = ρ l log c, β 0 = log c, β 1 = log c. E 0 [N] lim c 0 log c 1 E 1 [N] Dtot 0 + M 0 and lim c 0 log c 1 Dtot 1 + M 1 Gaussian MAC noise at FC µ chosen appropriately lim c 0 P FA = 0 and lim c log c c 0 P MD c log c = 0

75 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}] L L Dtot 0 = D(f 0,l f 1,l ), Dtot 1 = D(f 1,l f 0,l ), r l = D(f 0,l f 1,l ), ρ Dtot 0 l = D(f 1,l f 0,l ) Dtot 1 l=1 Theorem l=1 γ 0,l = r l log c,γ 1,l = ρ l log c, β 0 = log c, β 1 = log c. E 0 [N] lim c 0 log c 1 E 1 [N] Dtot 0 + M 0 and lim c 0 log c 1 Dtot 1 + M 1 Gaussian MAC noise at FC µ chosen appropriately lim c 0 P FA = 0 and lim c log c c 0 R c (δbayes lim ) = 1 c 0 R c (δ) P MD c log c = 0

76 Asymptotic Properties of DualSPRT R c(δ) = π[c E 0(N) + W 0P 0{reject H 0}] + (1 π)[c E 1(N) + W 1P 1{rejectH 1}] L L Dtot 0 = D(f 0,l f 1,l ), Dtot 1 = D(f 1,l f 0,l ), r l = D(f 0,l f 1,l ), ρ Dtot 0 l = D(f 1,l f 0,l ) Dtot 1 l=1 Theorem l=1 γ 0,l = r l log c,γ 1,l = ρ l log c, β 0 = log c, β 1 = log c. E 0 [N] lim c 0 log c 1 E 1 [N] Dtot 0 + M 0 and lim c 0 log c 1 Dtot 1 + M 1 Gaussian MAC noise at FC µ chosen appropriately lim c 0 P FA = 0 and lim c log c c 0 P MD c log c = 0 R c (δbayes lim ) R c (δcent. = 1 and lim ) = 1 c 0 R c (δ) c 0 R c (δ)

77 Modification to DualSPRT: SPRT-CSPRT Algorithm Using SPRT at Fusion center is not optimal since the optimality of SPRT is known for i.i.d. observations only.

78 Modification to DualSPRT: SPRT-CSPRT Algorithm Using SPRT at Fusion center is not optimal since the optimality of SPRT is known for i.i.d. observations only. What is the best test at FC?

79 Modification to DualSPRT: SPRT-CSPRT Algorithm Using SPRT at Fusion center is not optimal since the optimality of SPRT is known for i.i.d. observations only. What is the best test at FC? Sequential change detection formulation µ k = E[Y k ] E[Yk] Change point t1 k

80 Modification to DualSPRT: SPRT-CSPRT Algorithm Using SPRT at Fusion center is not optimal since the optimality of SPRT is known for i.i.d. observations only. What is the best test at FC? Sequential change detection formulation µ k = E[Y k ] E[Yk] Change point t1 k Reducing the false alarms caused by FC MAC noise before t 1.

81 SPRT-CSPRT: (inspired from CUSUM) FC algorithm, ( Fk 1 = Fk log g ) + ( µ 1 (Y k ), Fk 0 = Fk log g ) z(y k ) g z (Y k ) g µ0 (Y k )

82 SPRT-CSPRT: (inspired from CUSUM) FC algorithm, ( Fk 1 = Fk log g ) + ( µ 1 (Y k ), Fk 0 = Fk log g ) z(y k ) g z (Y k ) g µ0 (Y k ) FC decides at N = inf{k : F 1 k β 1 or F 0 k β 0}, H 1 if F 1 N β 1, H 0 if F 0 N β 0

83 SPRT-CSPRT: (inspired from CUSUM) FC algorithm, ( Fk 1 = Fk log g ) + ( µ 1 (Y k ), Fk 0 = Fk log g ) z(y k ) g z (Y k ) g µ0 (Y k ) FC decides at N = inf{k : F 1 k β 1 or F 0 k β 0}, H 1 if F 1 N β 1, H 0 if F 0 N β 0 [ ] µ k = E[Y k ], E µk log gµ 1 (Y k ) = D(g µk g Z ) D(g µk g µ1 ) g Z (Y k )

84 SPRT-CSPRT: (inspired from CUSUM) FC algorithm, ( Fk 1 = Fk log g ) + ( µ 1 (Y k ), Fk 0 = Fk log g ) z(y k ) g z (Y k ) g µ0 (Y k ) FC decides at N = inf{k : F 1 k β 1 or F 0 k β 0}, H 1 if F 1 N β 1, H 0 if F 0 N β 0 [ ] µ k = E[Y k ], E µk log gµ 1 (Y k ) = D(g µk g Z ) D(g µk g µ1 ) g Z (Y k ) Exp. drift of F 1 k +ve if D(g µ k g Z ) > D(g µk g µ1 ).

85 SPRT-CSPRT: (inspired from CUSUM) FC algorithm, ( Fk 1 = Fk log g ) + ( µ 1 (Y k ), Fk 0 = Fk log g ) z(y k ) g z (Y k ) g µ0 (Y k ) FC decides at N = inf{k : F 1 k β 1 or F 0 k β 0}, H 1 if F 1 N β 1, H 0 if F 0 N β 0 [ ] µ k = E[Y k ], E µk log gµ 1 (Y k ) = D(g µk g Z ) D(g µk g µ1 ) g Z (Y k ) Exp. drift of F 1 k +ve if D(g µ k g Z ) > D(g µk g µ1 ). Exp. drift of F 0 k -ve if D(g µ k g µ0 ) < D(g µk g Z ).

86 SPRT-CSPRT: (inspired from CUSUM) FC algorithm, ( Fk 1 = Fk log g ) + ( µ 1 (Y k ), Fk 0 = Fk log g ) z(y k ) g z (Y k ) g µ0 (Y k ) FC decides at N = inf{k : F 1 k β 1 or F 0 k β 0}, H 1 if F 1 N β 1, H 0 if F 0 N β 0 [ ] µ k = E[Y k ], E µk log gµ 1 (Y k ) = D(g µk g Z ) D(g µk g µ1 ) g Z (Y k ) Exp. drift of F 1 k +ve if D(g µ k g Z ) > D(g µk g µ1 ). Exp. drift of F 0 k -ve if D(g µ k g µ0 ) < D(g µk g Z ). Sample path argument SPRT sum CSPRT sum β 1 20 F k Time (k)

87 Modification of quantisation at SU s γ 1 +3Δ 1 γ 1 +2Δ 1 γ 1 +Δ 1 γ 1 b 1 1 b 1 2 b 1 3 b 1 4 W k b 1 1 <b 1 2< b 1 3< b 1 4 b 0 1 >b 0 2>b 0 3>b 0 4 Time γ 0 γ 0 -Δ 0 b 0 1 γ 0-2Δ 0 γ 0-3Δ 0 b 0 2 b 0 3 b 0 4

88 Performance Analysis of SPRT-CSPRT: P MD P H1 (MD) = P H1 (MD < t 1 ) + P H1 (t 1 MD t 2 ) +... First term dominates.

89 Performance Analysis of SPRT-CSPRT: P MD P H1 (MD) = P H1 (MD < t 1 ) + P H1 (t 1 MD t 2 ) +... First term dominates. P H1 (FA before t 1 ) = P(τ β k k < t 1 )P(t 1 > k) k=1 τ β = inf{k 1 : F 0 k β}

90 Performance Analysis of SPRT-CSPRT: P MD P H1 (MD) = P H1 (MD < t 1 ) + P H1 (t 1 MD t 2 ) +... First term dominates. P H1 (FA before t 1 ) = P(τ β k k < t 1 )P(t 1 > k) k=1 τ β = inf{k 1 : F 0 k β} lim P 0{τ β > x τ β < t 1 } = exp( λ β x), x > 0 β

91 Performance Analysis of SPRT-CSPRT: P MD P H1 (MD) = P H1 (MD < t 1 ) + P H1 (t 1 MD t 2 ) +... First term dominates. P H1 (FA before t 1 ) = P(τ β k k < t 1 )P(t 1 > k) k=1 τ β = inf{k 1 : F 0 k β} lim P 0{τ β > x τ β < t 1 } = exp( λ β x), x > 0 β P H1 (FA before t 1 ) L (1 e λβk ) (1 Φ τγ,l (k)) k=1 l=1

92 Comparison of Analysis with Simulations 5 Secondary nodes channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1 P MD Sim. P MD Anal. E DD Sim. E DD Anal

93 Comparison with asymptotically optimal tests 10 E DD DualSPRT DSPRT SPRT CSPRT Mei s SPRT P x 10 3 FA Mei s SPRT: Mei, Trans. Inf. Theory DSPRT: Fellouris and Moustakides, Trans. Inf. Theory 2011.

94 Different and unknown SNR s: GLR-SPRT SNR at local CR nodes Unknown

95 Different and unknown SNR s: GLR-SPRT SNR at local CR nodes Unknown Energy detector output is used as the observations to the local node SPRT s

96 Different and unknown SNR s: GLR-SPRT SNR at local CR nodes Unknown Energy detector output is used as the observations to the local node SPRT s Modelled as Gaussian mean change under low SNR s and large number of samples

97 Different and unknown SNR s: GLR-SPRT SNR at local CR nodes Unknown Energy detector output is used as the observations to the local node SPRT s Modelled as Gaussian mean change under low SNR s and large number of samples At SU l, H 0 : θ = θ 0 ; H 1 : θ θ 1. where θ 0 = 0 and θ 1 is appropriately chosen,

98 Different and unknown SNR s: GLR-SPRT SNR at local CR nodes Unknown Energy detector output is used as the observations to the local node SPRT s Modelled as Gaussian mean change under low SNR s and large number of samples At SU l, H 0 : θ = θ 0 ; H 1 : θ θ 1. where θ 0 = 0 and θ 1 is appropriately chosen, [ n W n,l = max log f ˆθ n (X k ) n f θ0 (X k ), log f ] ˆθ n (X k ), f θ1 (X k ) k=1 k=1

99 Different and unknown SNR s: GLR-SPRT SNR at local CR nodes Unknown Energy detector output is used as the observations to the local node SPRT s Modelled as Gaussian mean change under low SNR s and large number of samples At SU l, H 0 : θ = θ 0 ; H 1 : θ θ 1. where θ 0 = 0 and θ 1 is appropriately chosen, [ n W n,l = max log f ˆθ n (X k ) n f θ0 (X k ), log f ] ˆθ n (X k ), f θ1 (X k ) k=1 k=1 N = inf {n : W n,l g(cn)}, g(t) log(1/t) as t 0

100 Decide upon H 0 or H 1 as ˆθ N θ or ˆθ N θ where θ solves D(f θ f θ0 ) = D(f θ f θ1 ) D(f θ f λ ) = E θ [log[f θ (X )/f λ (X )]

101 Decide upon H 0 or H 1 as ˆθ N θ or ˆθ N θ where θ solves D(f θ f θ0 ) = D(f θ f θ1 ) D(f θ f λ ) = E θ [log[f θ (X )/f λ (X )] Similar to GLR test in Neyman-Pearson approach in FSS tests Diminishing uncertainty of ˆθ n as an estimate of θ with n is incorporated into the varying stopping boundary g(cn) Asymptotically optimal over a broad range of θ as c 0

102 Comparison between DualSPRT and GLRSPRT 5 Secondary nodes

103 Comparison between DualSPRT and GLRSPRT 5 Secondary nodes channel gains (0, -1.5, -2.5, -4 and -6 db)

104 Comparison between DualSPRT and GLRSPRT 5 Secondary nodes channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1

105 Comparison between DualSPRT and GLRSPRT 5 Secondary nodes channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1 hyp E DD P E = 0.1 P E = 0.05 P E = 0.01 H1 DualSPRT H1 GLRSPRT H0 DualSPRT H0 GLRSPRT

106 Rayleigh Slow fading case N is much less than the coherence time of the channel

107 Rayleigh Slow fading case N is much less than the coherence time of the channel Fading gain is unknown to the CR nodes

108 Rayleigh Slow fading case N is much less than the coherence time of the channel Fading gain is unknown to the CR nodes In case of Rayleigh fading, the received signal power, P l is exponentially distributed

109 Rayleigh Slow fading case N is much less than the coherence time of the channel Fading gain is unknown to the CR nodes In case of Rayleigh fading, the received signal power, P l is exponentially distributed H 0 : f 0,l N (0, σ 2 ); H 1 : f 1,l N (θ, σ 2 ) where θ is exponential distributed r.v which reflects the unknown channel gain.

110 Simulation results No. of nodes :5 σ 2 = 1, θ = exp(1), Var(Z k ) = 1

111 Simulation results No. of nodes :5 σ 2 = 1, θ = exp(1), Var(Z k ) = 1 E DD P FA = 0.1 P FA = 0.05 P FA = 0.01 DualSPRT GLRSPRT Table: slow-fading between primary and secondary user under H0

112 Simulation results No. of nodes :5 σ 2 = 1, θ = exp(1), Var(Z k ) = 1 E DD P FA = 0.1 P FA = 0.05 P FA = 0.01 DualSPRT GLRSPRT Table: slow-fading between primary and secondary user under H0 E DD P MD = 0.1 P MD = 0.08 P MD = 0.06 DualSPRT GLRSPRT Table: slow-fading between primary and secondary user under H1

113 Comparison between SPRT-CSPRT and GLR-CSPRT 5 Secondary nodes Unknown SNR: channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1

114 Comparison between SPRT-CSPRT and GLR-CSPRT 5 Secondary nodes Unknown SNR: channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1 Hyp E DD P E = 0.1 P E = 0.05 P E = 0.01 H 1 SPRT-CSPRT H 1 GLR-CSPRT H 0 SPRT-CSPRT H 0 GLR-CSPRT

115 Comparison between SPRT-CSPRT and GLR-CSPRT 5 Secondary nodes Unknown SNR: channel gains (0, -1.5, -2.5, -4 and -6 db) f 0 N (0, 1) and f 1,l N (θ l, 1), Var(Z k ) = 1 Hyp E DD P E = 0.1 P E = 0.05 P E = 0.01 H 1 SPRT-CSPRT H 1 GLR-CSPRT H 0 SPRT-CSPRT H 0 GLR-CSPRT Rayleigh Slow fading: σ 2 = 1, θ = exp(1), Var(Z k ) = 1 Hyp E DD P E = 0.1 P E = 0.07 P E = 0.04 H 1 SPRT-CSPRT H 1 GLR-CSPRT H 0 SPRT-CSPRT H 0 GLR-CSPRT

116 Universal Sequential Hypothesis Testing

117 Universal sequential hypothesis testing problem Model for Single Cognitive Radio Nonparametric or universal setup: Noise statistics under no PU transmission is fully known Transmit power, channel gains, modulation schemes etc. of PU transmissions is not available (SNR uncertainty). i.i.d. observations X i, i = 1, 2,... H 0 : X i P 0 ( p.d.f. f 0 ) known ; H 1 : X i P 1 ( p.d.f. f 1 ) unknown

118 Discrete alphabet for P 0 and P 1 SPRT: P 0 and P 1 fully known, for γ 0, γ 1 > 0

119 Discrete alphabet for P 0 and P 1 SPRT: P 0 and P 1 fully known, for γ 0, γ 1 > 0 N = inf{n : W n = n k=1 log P 1(X k ) P 0 (X k ) / ( γ 0, γ 1 )},

120 Discrete alphabet for P 0 and P 1 SPRT: P 0 and P 1 fully known, for γ 0, γ 1 > 0 N = inf{n : W n = n k=1 log P 1(X k ) P 0 (X k ) / ( γ 0, γ 1 )}, δ = H 1 if W N γ 1 ; H 0 if W N γ 0

121 Discrete alphabet for P 0 and P 1 SPRT: P 0 and P 1 fully known, for γ 0, γ 1 > 0 N = inf{n : W n = n k=1 log P 1(X k ) P 0 (X k ) / ( γ 0, γ 1 )}, δ = H 1 if W N γ 1 ; H 0 if W N γ 0 P 0 is known; P 1 is not known.

122 Discrete alphabet for P 0 and P 1 SPRT: P 0 and P 1 fully known, for γ 0, γ 1 > 0 N = inf{n : W n = n k=1 log P 1(X k ) P 0 (X k ) / ( γ 0, γ 1 )}, δ = H 1 if W N γ 1 ; H 0 if W N γ 0 P 0 is known; P 1 is not known. Replace W n by Ŵ n = L n (X n 1 ) log P 0 (X n 1 ) n λ 2, λ > 0. L n (X n 1 )=Codelength of a universal lossless source code for X n 1

123 Discrete alphabet case Discussion of the test By Shanon-Macmillan Thorem, for any stationary ergodic source lim n n 1 log P(X n 1 ) = H(X ) a.s. ( H(X ) is the entropy rate).

124 Discrete alphabet case Discussion of the test By Shanon-Macmillan Thorem, for any stationary ergodic source lim n n 1 log P(X n 1 ) = H(X ) a.s. ( H(X ) is the entropy rate). Consider universal codes which satisfy lim n n 1 L n = H(X ) a.s. atleast for i.i.d. sources.

125 Discrete alphabet case Discussion of the test By Shanon-Macmillan Thorem, for any stationary ergodic source lim n n 1 log P(X n 1 ) = H(X ) a.s. ( H(X ) is the entropy rate). Consider universal codes which satisfy lim n n 1 L n = H(X ) a.s. atleast for i.i.d. sources. Under H 1 : E1 [( log P 0 (X n 1 ))] = nh 1(X ) + nd(p 1 P 0 )

126 Discrete alphabet case Discussion of the test By Shanon-Macmillan Thorem, for any stationary ergodic source lim n n 1 log P(X n 1 ) = H(X ) a.s. ( H(X ) is the entropy rate). Consider universal codes which satisfy lim n n 1 L n = H(X ) a.s. atleast for i.i.d. sources. Under H 1 : E1 [( log P 0 (X1 n))] = nh 1(X ) + nd(p 1 P 0 ) L(X1 n) nh 1(X ) for large n

127 Discrete alphabet case Discussion of the test By Shanon-Macmillan Thorem, for any stationary ergodic source lim n n 1 log P(X n 1 ) = H(X ) a.s. ( H(X ) is the entropy rate). Consider universal codes which satisfy lim n n 1 L n = H(X ) a.s. atleast for i.i.d. sources. Under H 1 : E1 [( log P 0 (X1 n))] = nh 1(X ) + nd(p 1 P 0 ) L(X1 n) nh 1(X ) for large n Thus average drift under H 1 : D(P 1 P 0 ) λ/2

128 Discrete alphabet case Discussion of the test By Shanon-Macmillan Thorem, for any stationary ergodic source lim n n 1 log P(X n 1 ) = H(X ) a.s. ( H(X ) is the entropy rate). Consider universal codes which satisfy lim n n 1 L n = H(X ) a.s. atleast for i.i.d. sources. Under H 1 : E1 [( log P 0 (X1 n))] = nh 1(X ) + nd(p 1 P 0 ) L(X1 n) nh 1(X ) for large n Thus average drift under H 1 : D(P 1 P 0 ) λ/2 Average drift under H 0 : λ/2

129 Discrete alphabet case Discussion of the test By Shanon-Macmillan Thorem, for any stationary ergodic source lim n n 1 log P(X n 1 ) = H(X ) a.s. ( H(X ) is the entropy rate). Consider universal codes which satisfy lim n n 1 L n = H(X ) a.s. atleast for i.i.d. sources. Under H 1 : E1 [( log P 0 (X1 n))] = nh 1(X ) + nd(p 1 P 0 ) L(X1 n) nh 1(X ) for large n Thus average drift under H 1 : D(P 1 P 0 ) λ/2 Average drift under H 0 : λ/2 Thus consider P 1 belonging to the class C = {P 1 : D(P 1, P 0 ) λ}.

130 Discrete alphabet case Asymptotic Properties (A): stronger version of pointwise universality Theorem (1) P 0 (N < ) = 1 and P 1 (N < ) = 1.

131 Discrete alphabet case Asymptotic Properties (A): stronger version of pointwise universality Theorem (1) P 0 (N < ) = 1 and P 1 (N < ) = 1. (2) P FA = P0 (ŴN γ 1 ) exp( γ 1 ).

132 Discrete alphabet case Asymptotic Properties (A): stronger version of pointwise universality Theorem (1) P 0 (N < ) = 1 and P 1 (N < ) = 1. (2) P FA = P0 (ŴN γ 1 ) exp( γ 1 ). (3) Under (A), P MD = P1 (ŴN γ 0 ) = O(exp( γ 0 s)), s > 0..

133 Discrete alphabet case Asymptotic Properties (A): stronger version of pointwise universality Theorem (1) P 0 (N < ) = 1 and P 1 (N < ) = 1. (2) P FA = P0 (ŴN γ 1 ) exp( γ 1 ). (3) Under (A), P MD = P1 (ŴN γ 0 ) = O(exp( γ 0 s)), s > 0.. (4) N P 0 a.s. 2 log γ 0 γ 0,γ 1 λ ; N log γ 1 P 1, a.s. γ 0,γ 1 1 D(P 1 P 0 ) λ/2

134 Discrete alphabet case Asymptotic Properties (A): stronger version of pointwise universality Theorem (1) P 0 (N < ) = 1 and P 1 (N < ) = 1. (2) P FA = P0 (ŴN γ 1 ) exp( γ 1 ). (3) Under (A), P MD = P1 (ŴN γ 0 ) = O(exp( γ 0 s)), s > 0.. (4) N P 0 a.s. 2 log γ 0 γ 0,γ 1 λ ; N log γ 1 P 1, a.s. γ 0,γ 1 1 D(P 1 P 0 ) λ/2 (5) Under (A), if E 1 [(log P 1 (X 1 )) p+1 ] < & E 1 [(log P 0 (X 1 )) p+1 ] < for some p 1, then for all 0 < q p, E 0 [N q ( ) ( q ] 2 E 1 [N q ] 1 log γ 0 q, γ 0,γ 1 λ log γ 1 q γ 0,γ 1 D(P 1 P 0 ) λ 2 ) q

135 Continuous alphabet for P 0 and P 1 Need to estimate n k=1 log f 1(X k )

136 Continuous alphabet for P 0 and P 1 Need to estimate n k=1 log f 1(X k ) If E[log f 1 (X k )] < then by SLLN n 1 n log f 1 (X k ) h(x 1 ) a.s. k=1

137 Continuous alphabet for P 0 and P 1 Need to estimate n k=1 log f 1(X k ) If E[log f 1 (X k )] < then by SLLN n 1 n log f 1 (X k ) h(x 1 ) a.s. k=1 X k X k = [X k/ ] ;

138 Continuous alphabet for P 0 and P 1 Need to estimate n k=1 log f 1(X k ) If E[log f 1 (X k )] < then by SLLN n 1 n log f 1 (X k ) h(x 1 ) a.s. k=1 X k Xk = [X k/ ] ; Use Universal lossless coding on X1, X 2,..., X n

139 Continuous alphabet for P 0 and P 1 Need to estimate n k=1 log f 1(X k ) If E[log f 1 (X k )] < then by SLLN n 1 n log f 1 (X k ) h(x 1 ) a.s. k=1 X k Xk = [X k/ ] ; Use Universal lossless coding on X1, X 2,..., X n H(X1 ) + log h(x 1) as 0.

140 Continuous alphabet for P 0 and P 1 Need to estimate n k=1 log f 1(X k ) If E[log f 1 (X k )] < then by SLLN n 1 n log f 1 (X k ) h(x 1 ) a.s. k=1 X k Xk = [X k/ ] ; Use Universal lossless coding on X1, X 2,..., X n H(X1 ) + log h(x 1) as 0. Test is modified to W n = L n (X 1:n) n log n log f 0 (X k ) n λ 2 k=1

141 Continuous alphabet for P 0 and P 1 Need to estimate n k=1 log f 1(X k ) If E[log f 1 (X k )] < then by SLLN n 1 n log f 1 (X k ) h(x 1 ) a.s. k=1 X k Xk = [X k/ ] ; Use Universal lossless coding on X1, X 2,..., X n H(X1 ) + log h(x 1) as 0. Test is modified to W n = L n (X 1:n) n log n log f 0 (X k ) n λ 2 k=1 f 1 C = {f 1 : D(f 1, f 0 ) λ}.

142 Continuous alphabet case Why Uniform Quantization?

143 Continuous alphabet case Why Uniform Quantization? Uniform scalar quantization with optimal source coding is optimal at high rates.

144 Continuous alphabet case Why Uniform Quantization? Uniform scalar quantization with optimal source coding is optimal at high rates. An adaptive uniform quantizer makes {X n } non i.i.d. L n unable to learn the underlying distribution in such scenario.

145 Continuous alphabet case Why Uniform Quantization? Uniform scalar quantization with optimal source coding is optimal at high rates. An adaptive uniform quantizer makes {X n } non i.i.d. L n unable to learn the underlying distribution in such scenario. Non-uniform partitions with j in j th bin and with p.m.f p j require knowledge of p j which is unknown under H 1.

146 Continuous alphabet case Why Uniform Quantization? Uniform scalar quantization with optimal source coding is optimal at high rates. An adaptive uniform quantizer makes {X n } non i.i.d. L n unable to learn the underlying distribution in such scenario. Non-uniform partitions with j in j th bin and with p.m.f p j require knowledge of p j which is unknown under H 1. f 0 (X i ) p 0 (X i ). W n = L n (Xi ) n j=1 log p 0(Xi ) nλ/2.

147 Continuous alphabet case Why Uniform Quantization? Uniform scalar quantization with optimal source coding is optimal at high rates. An adaptive uniform quantizer makes {X n } non i.i.d. L n unable to learn the underlying distribution in such scenario. Non-uniform partitions with j in j th bin and with p.m.f p j require knowledge of p j which is unknown under H 1. j=1 log p 0(Xi ) nλ/2. Range of the quantization: f 1 s tail probabilities less than a small specific value at a fixed boundary f 0 (X i ) p 0 (X i ). W n = L n (Xi ) n

148 LZSLRT algorithm Use Lempel-Ziv incremental parsing technique (LZ78).

149 LZSLRT algorithm Use Lempel-Ziv incremental parsing technique (LZ78). Added a correction term nɛ n in W n

150 LZSLRT algorithm Use Lempel-Ziv incremental parsing technique (LZ78). Added a correction term nɛ n in W n ɛ n = C ( 1 log log n + + log n n C depends on size of the quantized alphabet. ) log log n. log n

151 LZSLRT algorithm Use Lempel-Ziv incremental parsing technique (LZ78). Added a correction term nɛ n in W n ɛ n = C ( 1 log log n + + log n n C depends on size of the quantized alphabet. ) log log n. log n LZSLRT performance is close to the nearly optimal parametric GLR sequential tests and is better for some class of distributions (e.g. Pareto)

152 KTSLRT algorithm L n (X 1:n ) from combined use of Krichevsky-Trofimov estimator and the Arithmetic Encoder

153 KTSLRT algorithm L n (X1:n ) from combined use of Krichevsky-Trofimov estimator and the Arithmetic Encoder Estimate source distribution via K-T estimator for a finite alphabet source as, P c (x n 1 ) = n t=1 v(x t /x t 1 1 ) t 1 + A 2,

154 KTSLRT algorithm L n (X1:n ) from combined use of Krichevsky-Trofimov estimator and the Arithmetic Encoder Estimate source distribution via K-T estimator for a finite alphabet source as, P c (x n 1 ) = n t=1 v(x t /x t 1 1 ) t 1 + A 2, Using AE with above estimate: L n (x n 1 ) < log P c(x n 1 ) + 2

155 KTSLRT algorithm L n (X1:n ) from combined use of Krichevsky-Trofimov estimator and the Arithmetic Encoder Estimate source distribution via K-T estimator for a finite alphabet source as, P c (x n 1 ) = n t=1 v(x t /x t 1 1 ) t 1 + A 2, Using AE with above estimate: L n (x n 1 ) < log P c(x n 1 ) + 2 Universal codes defined via the K-T estimator and AE are nearly optimal

156 Performance Comparison LZLRT E DD = 0.5 E H1 [N] E H0 [N] versus P E = 0.5 P FA P MD

157 Performance Comparison LZLRT E DD = 0.5 E H1 [N] E H0 [N] versus P E = 0.5 P FA P MD E DD P E = 0.05 P E = 0.01 P E = SPRT GLR-Lai LZSLRT Table: f 0 N (0, 5) and f 0 N (3, 5)

158 Performance Comparison LZLRT E DD = 0.5 E H1 [N] E H0 [N] versus P E = 0.5 P FA P MD E DD P E = 0.05 P E = 0.01 P E = SPRT GLR-Lai LZSLRT Table: f 0 N (0, 5) and f 0 N (3, 5) E DD P E = 0.05 P E = 0.01 P E = SPRT GLR-Lai LZSLRT Table: f 0 P(10, 2) and f 0 P(3, 2)

159 Performance Comparison KTSLRT Gaussian case. f 1 N (0, 5) and f 0 N (0, 1), 8 bit uniform quantizer KTSLRT S=1 KTSLRT S=0 LZSLRT 30 E DD P E

160 Performance Comparison Lognormal Distribution and Pareto Distribution examples f 1 lnn (3, 3) and f 0 lnn (0, 3) KTSLRT S=1 LZSLRT E DD P E

161 Performance Comparison Lognormal Distribution and Pareto Distribution examples f 1 lnn (3, 3) and f 0 lnn (0, 3) f 1 P(3, 2) and f 0 P(10, 2), support set (2, 10) KTSLRT S=1 LZSLRT KTSLRT S=1 LZSLRT E DD 10 8 E DD P E P E

162 Performance Comparison Gaussian case (f 1 N (0, 5) and f 0 N (0, 1)) with different estimators

163 Performance Comparison Gaussian case (f 1 N (0, 5) and f 0 N (0, 1)) with different estimators 1-NN differential entropy estimator is, γ = Euler-Mascheroni constant ĥ n = 1 n log ρ(i) + log(n 1) + γ + 1, ρ(i) = min n X i X j j:1 j n,j i i=1

164 Performance Comparison Gaussian case (f 1 N (0, 5) and f 0 N (0, 1)) with different estimators 1-NN differential entropy estimator is, γ = Euler-Mascheroni constant ĥ n = 1 n log ρ(i) + log(n 1) + γ + 1, ρ(i) = min n X i X j j:1 j n,j i i=1 Kernel density estimator at a point z is ˆf n (z) = 1 n ( ) z Xi K, k: kernel and w n : bandwidth w n w n i=1

165 Performance Comparison Gaussian case (f 1 N (0, 5) and f 0 N (0, 1)) with different estimators 1-NN differential entropy estimator is, γ = Euler-Mascheroni constant ĥ n = 1 n log ρ(i) + log(n 1) + γ + 1, ρ(i) = min n X i X j j:1 j n,j i i=1 Kernel density estimator at a point z is ˆf n (z) = 1 n ( ) z Xi K, k: kernel and w n : bandwidth w n w n i=1 KTSLRT S=1 Test with Kernel Denisty Estr. Test with 1NN Differential Entropy Estr. E DD P E

166 Performance Comparison Comparison with Hoeffding Test: P 0 B(8, 0.2), P 1 B(8, 0.5)

167 Performance Comparison Comparison with Hoeffding Test: P 0 B(8, 0.2), P 1 B(8, 0.5) Hoeffding test: Asymptotically optimal universal fixed sample size test for finite alphabet sources

168 Performance Comparison Comparison with Hoeffding Test: P 0 B(8, 0.2), P 1 B(8, 0.5) Hoeffding test: Asymptotically optimal universal fixed sample size test for finite alphabet sources δ FSS = I{D(Γ n P 0 ) η}

169 Performance Comparison Comparison with Hoeffding Test: P 0 B(8, 0.2), P 1 B(8, 0.5) Hoeffding test: Asymptotically optimal universal fixed sample size test for finite alphabet sources δ FSS = I{D(Γ n P 0 ) η} KT estimator with AE Hoeffding test 20 E DD P E

170 Decentralized case Model and Algorithm Sequential+Universal+Cooperative

171 Decentralized case Model and Algorithm Sequential+Universal+Cooperative observations are i.i.d. at any CR and independent across CRs.

172 Decentralized case Model and Algorithm Sequential+Universal+Cooperative observations are i.i.d. at any CR and independent across CRs. Algorithm: at SU s:, at FC:

173 Decentralized case Model and Algorithm Sequential+Universal+Cooperative observations are i.i.d. at any CR and independent across CRs. Algorithm: at SU s: LZSLRT/KTSLRT, at FC: SPRT

174 Decentralized case Model and Algorithm Sequential+Universal+Cooperative observations are i.i.d. at any CR and independent across CRs. Algorithm: at SU s: LZSLRT/KTSLRT, at FC: SPRT noisy MAC between SUs and FC FC SPRT: binary hypothesis testing of g µ1 vs g µ0

175 Decentralized case Performance Comparison b 1 = 1, b 0 = 1, I = 2, L = 5 and Z k N (0, 1)

176 Decentralized case Performance Comparison b 1 = 1, b 0 = 1, I = 2, L = 5 and Z k N (0, 1) DualSPRT KTSLRT SPRT S=1 LZSLRT SPRT E DD P E f 0,l N (0, 1) and f 1,l N (0, 5), for 1 l L.

177 Decentralized case Performance Comparison b 1 = 1, b 0 = 1, I = 2, L = 5 and Z k N (0, 1) E DD DualSPRT KTSLRT SPRT S=1 LZSLRT SPRT P E E DD DualSPRT KTSLRT SPRT S=1 LZSLRT SPRT P E f 0,l N (0, 1) and f 1,l N (0, 5), for 1 l L. f 0,l P(10, 2) and f 1,l P(3, 2), for 1 l L.

178 Decentralized case Performance Comparison b 1 = 1, b 0 = 1, I = 2, L = 5 and Z k N (0, 1) E DD DualSPRT KTSLRT SPRT S=1 LZSLRT SPRT P E E DD DualSPRT KTSLRT SPRT S=1 LZSLRT SPRT P E f 0,l N (0, 1) and f 1,l N (0, 5), for 1 l L. f 0,l P(10, 2) and f 1,l P(3, 2), for 1 l L. Analysis using Perturbed Random Walk theory Ŵ n = S n + ξ n, ξ n /n 0 a.s.

179 Multihypothesis Decentralized Sequential Testing (distributions completely known)

180 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1

181 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = (X 1 )

182 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 )

183 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 ) [ min E i log f k ] (X 1 ) j k f j = (X 1 )

184 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 ) [ min E i log f k ] (X 1 ) j k f j = (X 1 ) { < 0 when k i

185 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 ) [ min E i log f k ] (X 1 ) j k f j = (X 1 ) { < 0 when k i > 0 when k = i

186 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 ) [ min E i log f k ] (X 1 ) j k f j = (X 1 ) { < 0 when k i > 0 when k = i At local node l, {W k,j n,l, 0 k, j M 1}

187 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 ) [ min E i log f k ] (X 1 ) j k f j = (X 1 ) { < 0 when k i > 0 when k = i At local node l, {W k,j n,l, 0 k, j M 1} W k,j n,l = W k,j n 1,l + log f l k (X n,l ) f j l (X n,l), W k,j 0,l = 0

188 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 ) [ min E i log f k ] (X 1 ) j k f j = (X 1 ) { < 0 when k i > 0 when k = i At local node l, {W k,j n,l, 0 k, j M 1} W k,j n,l = W k,j n 1,l + log f l k (X n,l ) f j l (X n,l), W k,j 0,l = 0 N l = inf{n : W k,j n,l > A for all j k and some k}

189 Multihypothesis Decentralized Sequential Testing: Algorithm-1 (DMSLRT-1) M Hypothesis (M > 2), H i : X n f i, i = 0,..., M 1 [ E i log f k ] (X 1 ) f j = D(f i f j ) D(f i f k ) (X 1 ) [ min E i log f k ] (X 1 ) j k f j = (X 1 ) { < 0 when k i > 0 when k = i At local node l, {W k,j n,l, 0 k, j M 1} W k,j n,l = W k,j n 1,l + log f l k (X n,l ) f j l (X n,l), W k,j 0,l = 0 N l = inf{n : W k,j n,l > A for all j k and some k} or N l = inf{n : max min W k,j k j k n,l > A}

190 Algorithm-1 (DMSLRT-1) Contd. Use reflected random walks at local nodes, max(w n,l, 0)

191 Algorithm-1 (DMSLRT-1) Contd. Use reflected random walks at local nodes, max(w n,l, 0) Decision by node l = H i At time k N l, node l transmits b i

192 Algorithm-1 (DMSLRT-1) Contd. Use reflected random walks at local nodes, max(w n,l, 0) Decision by node l = H i At time k N l, node l transmits b i Instead of physical layer fusion, TDMA used.

193 Algorithm-1 (DMSLRT-1) Contd. Use reflected random walks at local nodes, max(w n,l, 0) Decision by node l = H i At time k N l, node l transmits b i Instead of physical layer fusion, TDMA used. FC uses the same test with hypotheses G m : Y k f m FC = N (b m, σ 2 )

194 Numerical results H m : X k,l N (m, 1), m=0,..., 4, No of local nodes=5 E DD DMSLRT 1 DualTest D1 MSLRT 1:Test D1 Test D1:MSLRT P FA Figure: Comparison among different Multihypothesis schemes

195 Algorithm-2 (DMSLRT-2) Reduce false alarms caused by Gaussian noise before first transmission from local nodes.

196 Algorithm-2 (DMSLRT-2) Reduce false alarms caused by Gaussian noise before first transmission from local nodes. FC statistic is modified as ( Fn i,j i,0 = F n F n j,0 i,0, where F n = max F i,0 n 1 + log f FC i (Y ) n) f Z (Y n ), 0.

197 Algorithm-2 (DMSLRT-2) Reduce false alarms caused by Gaussian noise before first transmission from local nodes. FC statistic is modified as ( Fn i,j i,0 = F n F n j,0 i,0, where F n = max F i,0 n 1 + log f FC i (Y ) n) f Z (Y n ), 0. Expected drift of F i,0 n > 0 only when E[Y n ] > b i /2

198 Algorithm-2 (DMSLRT-2) Reduce false alarms caused by Gaussian noise before first transmission from local nodes. FC statistic is modified as ( Fn i,j i,0 = F n F n j,0 i,0, where F n = max F i,0 n 1 + log f FC i (Y ) n) f Z (Y n ), 0. i,0 Expected drift of F n > 0 only when E[Y n ] > b i /2 Positive b i s make E[Fn i,j ] negligible before first transmission

199 Numerical results SNRs (-10 db, -6 db, 0 db and 6 db), PU with SNR -10 db uses the channel DMSLRT 2 DMSLRT E DD P FA Figure: Comparison between MDSLRT-1 and MDSLRT-2

200 Analysis of Algorithm-1(DMSLRT-1) E DD Analysis at SU: Dominant event-reflected random walk with minimum positive expected drift

201 Analysis of Algorithm-1(DMSLRT-1) E DD Analysis at SU: Dominant event-reflected random walk with minimum positive expected drift E l DD = E i[n l ] A min j i D(f i l f j l )

202 Analysis of Algorithm-1(DMSLRT-1) E DD Analysis at SU: Dominant event-reflected random walk with minimum positive expected drift E l DD = E i[n l ] A min j i D(f i l f j l ) Nonlinear renewal theory to take care of overshoots

203 Analysis of Algorithm-1(DMSLRT-1) E DD Analysis at SU: Dominant event-reflected random walk with minimum positive expected drift E l DD = E i[n l ] A min j i D(f i l f j l ) Nonlinear renewal theory to take care of overshoots P FA Analysis at SU: Dominant event in {W k,j n,l, k i}-when the expected drift is most negative.

204 Analysis of Algorithm-1(DMSLRT-1) E DD Analysis at SU: Dominant event-reflected random walk with minimum positive expected drift E l DD = E i[n l ] A min j i D(f i l f j l ) Nonlinear renewal theory to take care of overshoots P FA Analysis at SU: Dominant event in {W k,j n,l, k i}-when the expected drift is most negative. N k,j l = inf{n : W k,j n,l ĵ = argmin j i D(f i l f j l ) > A} ; P l FA P(min k i N k,i l < N i,ĵ l )

205 Analysis of Algorithm-1(DMSLRT-1) E DD Analysis at SU: Dominant event-reflected random walk with minimum positive expected drift E l DD = E i[n l ] A min j i D(f i l f j l ) Nonlinear renewal theory to take care of overshoots P FA Analysis at SU: Dominant event in {W k,j n,l, k i}-when the expected drift is most negative. N k,j l = inf{n : W k,j n,l ĵ = argmin j i D(f i l f j l ) > A} ; P l FA P(min k i N k,i l < N i,ĵ l ) E DD Analysis of DMSLRT-1: E DD E i [max l N l ] + E i [N FC ]