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  • 1. Fundamental Limitsof Cognitive Radios<br />Goochul Chung<br />
  • 2. Outline<br />Motivation<br />Introduction<br />Part I ( Overlay Cognitive Radio )<br /><ul><li>Capacity Analysis of Overlay Cognitive Radio </li></ul>Part II ( Interweave Cognitive Radio )<br /><ul><li>Resource Allocation</li></ul>Part III ( Sensing in Interweave Cognitive Radio )<br /><ul><li>Sensing Strategy in Cooperative Sensing</li></li></ul><li>Frequency Allocation<br />Legitimate Radio<br /><ul><li>Licensed Radio with Primary Use of the Channel
  • 3. Stringent Regulation on Interference to Legitimate Radio</li></ul>Interference<br /><ul><li>Interference to Legitimate Radio
  • 4. Active Legitimate Radio
  • 5. Transmission over Legitimate Channel (Without Cooperation)</li></ul>Legitimate Radio<br />Secondary Network<br />
  • 6.
  • 7. Motivation for Cognitive Radio<br />Scarcity of Channel<br /><ul><li>Almost all wireless channels have licensed user with utmost privilege
  • 8. Constraint for new wireless applications</li></ul>Cognitive Radio<br /><ul><li>Possess Cognitive Information about Legitimate User
  • 9. Interference Avoidance
  • 10. Compensation</li></ul> Use legitimate channel without performance degradation of legitimate user<br /><ul><li>New Wireless Application with Cognitive Radio can Share the Channel with Legitimate User</li></li></ul><li>Cognitive Radio Type<br />Underlay Cognitive Radio<br /><ul><li>Cognitive Information
  • 11. Interference Level ( Channel Gain )
  • 12. Simultaneous Transmission with Limited Interference</li></ul>Interweave Cognitive Radio<br /><ul><li>Cognitive Information
  • 13. Channel Availability
  • 14. No Simultaneous Transmission </li></ul>Overlay Cognitive Radio<br /><ul><li>Cognitive Information
  • 15. Messages of Legitimate User
  • 16. Simultaneous Transmission </li></li></ul><li>Underlay Cognitive Radio<br />Underlay Cognitive Radio<br /><ul><li>Ultra Wide Band
  • 17. Spread its message over ultra wide band ( 3 – 10 GHz )
  • 18. Small Power
  • 19. Use small power to limit interference</li></ul>1<br />N<br />3 GHz<br />10 GHz<br />Legitimate Radio N<br />Legitimate Radio 1<br />Cognitive<br />Radio<br />
  • 20. Interweave Cognitive Radio<br />Interweave Cognitive Radio<br /><ul><li>No Simultaneous Transmission
  • 21. Sense if there is legitimate radio
  • 22. Transmit only if legitimate radio is not detected</li></ul>Legitimate Receiver<br />Legitimate Transmitter<br />Detection<br />No Transmission<br />Cognitive Transmitter<br />Cognitive Receiver<br />
  • 23. Overlay Cognitive Radio<br /><ul><li>Knowledge of Legitimate Users Message
  • 24. Obtain legitimate user’s message sets
  • 25. Unidirectional Cooperation</li></ul>Overlay Cognitive Radio<br />Legitimate Transmitter<br />Backbone Network<br />Legitimate Receiver<br />Unidirectional Cooperation<br />Cognitive Receiver<br />Cognitive Transmitter<br />
  • 26. Question<br />Cognitive Information Acquisition <br /><ul><li> Selective Information Acquisition
  • 27. Can information be efficiently collected to increase spectral efficiency? ( Channel Selection )
  • 28. Sensing
  • 29. How can sensing increase spectral efficiency with sensing accuracy?</li></ul>Utilization of Cognitive Information<br /><ul><li> Resource Allocation
  • 30. How can the resource (power) be allocated to increase average capacity?
  • 31. Coding
  • 32. What is the good coding strategy?</li></li></ul><li>Contribution <br />Capacity Analysis of Cognitive Radio<br /><ul><li>Information Theoretic Understanding on different types of Cognitive Radio
  • 33. Limits of Cognitive Radio
  • 34. Overlay Cognitive Radio
  • 35. Interweave Cognitive Radio
  • 36. Underlay Cognitive Radio</li></ul>System Design of Cognitive Radio<br /><ul><li>Build a Spectrally Efficient Cognitive Radio System
  • 37. Coding Strategy for Overlay Cognitive Radio
  • 38. Resource Allocation Algorithm for Interweave Cognitive Radio
  • 39. Sensing Strategy for Interweave Cognitive Radio </li></li></ul><li>Part ICapacity of Overlay Cognitive Radio<br />
  • 40. Previous Works<br />Fully Cognitive Radio<br /><ul><li> Cognitive of the entire message set of legitimate user
  • 41. Capacity of strong interference case (I. Maricet. al.)
  • 42. Capacity of general interference case (W. Wu et. al.) </li></ul>Partially Cognitive Radio<br /><ul><li>Cognitive of portion of the message set of legitimate user
  • 43. Capacity of strong interference case (I. Maricet. al.)</li></li></ul><li>Partially Cognitive Radio<br />Partially Cognitive Radio<br /><ul><li>Not fully cognitive of licensed user’s message sets
  • 44. : message set known to cognitive radio
  • 45. : message set unknown to cognitive radio</li></ul>Legitimate Transmitter<br />Cognitive Transmitter<br />Partially Cognitive Radio<br />Interference Channel<br />Fully Cognitive Radio<br />
  • 46. Motivation<br />Limited Cognitive Information<br /><ul><li>Practicality
  • 47. Knowledge of full non-causal message set is not guaranteed
  • 48. Generalization
  • 49. Partially Cognitive Radio generalize the channel model in between</li></ul> - Interference Channel<br /> - Fully Cognitive Radio Channel <br />Capacity Analysis <br /><ul><li>Information Theoretical Understanding
  • 50. Knowledge of limits of the channel
  • 51. Coding Design
  • 52. Knowledge of effective coding strategy</li></li></ul><li>System Model<br />Encoder1<br />Decoder1<br />Legitimate Transmitter<br />Legitimate Receiver<br />Encoder2<br />Decoder2<br />Cognitive Transmitter<br />Cognitive Receiver<br />Weak Interference<br /><ul><li>Weak Interference to Legitimate Receiver</li></ul>Rate Region<br /><ul><li> : Rate Triplet for </li></ul>Partially Cognitive Radio Constraint<br /><ul><li> : determines ratio of two rates
  • 53. : Fully Cognitive Radio Channel
  • 54. : Interference Channel ( Capacity : Still Open Problem )</li></li></ul><li>Key Feature<br />Common Message at the Cognitive Transmitter<br /><ul><li> : Detected only at Legitimate Receiver
  • 55. Unidirectional cooperation from the cognitive transmitter
  • 56. Side Information for Cognitive Radio </li></ul>Encoder1<br />Decoder1<br />Legitimate Transmitter<br />Legitimate Receiver<br />Encoder2<br />Decoder2<br />Cognitive Transmitter<br />Cognitive Receiver<br />
  • 57. Outer Bound (Discrete Memoryless Channel)<br />Outer bound of capacity region<br /><ul><li>Outer bound: Convex closure of the following inequality </li></ul>for the probability distribution <br />
  • 58. Outer Bound (Discrete Memoryless Channel)<br />Encoder1<br />Decoder1<br />Discrete Memoryless Channel<br />Legitimate Transmitter<br />Legitimate Receiver<br />Encoder2<br />Decoder2<br />Cognitive Receiver<br />Cognitive Transmitter<br />: Auxiliary Random Variable from<br />:Transmitter side information on<br /> : Decoded from at decoder1 <br />
  • 59. Outer Bound (Discrete Memoryless Channel)<br />Discrete Memoryless Channel<br />Encoder1<br />Decoder1<br />Legitimate Transmitter<br />Legitimate Receiver<br />Encoder2<br />Decoder2<br />Cognitive Transmitter<br />Cognitive Receiver<br /> : Auxiliary Random Variable from <br /> Independent with <br /> : Decoded from at decoder1 <br />
  • 60. Discrete Memoryless Channel<br />Encoder1<br />Decoder1<br />Legitimate Transmitter<br />Legitimate Receiver<br />Encoder2<br />Decoder2<br />Cognitive Transmitter<br />Cognitive Receiver<br />Outer Bound (Discrete Memoryless Channel)<br />: Encoded into<br />: Decoded from at decoder 2 <br />: Independent with <br />
  • 61. Discrete Memoryless Channel<br />Encoder1<br />Decoder1<br />Legitimate Transmitter<br />Legitimate Receiver<br />Encoder2<br />Decoder2<br />Cognitive Transmitter<br />Cognitive Receiver<br />Outer Bound (Discrete Memoryless Channel)<br />: Encoded into<br />: Decoded from at decoder 2 <br />: Independent with<br />: Given to encoder 2 as side information<br />
  • 62. Outer Bound (Gaussian Channel)<br />Gaussian Channel<br /><ul><li>Condition : weak interference to licensed receiver</li></ul>Encoder1<br />Decoder1<br />Legitimate Transmitter<br />Legitimate Receiver<br />Encoder2<br />Decoder2<br />Cognitive Transmitter<br />Cognitive Receiver<br />
  • 63. Outer Bound (Gaussian Channel)<br />Outer bound of capacity region<br /> : convex closure of set : Transmit powers <br />
  • 64. Outer Bound (Gaussian Channel)<br />Encoding With Power Split<br /><ul><li>Constraint on : Transmit cooperation with power split
  • 65. Same codeword for
  • 66. Constraint on : Interference free for transmission
  • 67. Interference cancellation at the transmitter </li></ul>Decoder1<br />Decoder2<br />Encoding With Power Split & Interference Cancellation<br />
  • 68. Achievable Scheme (General Idea)<br />Support Legitimate Transmission<br /><ul><li>Superposition coding with allocation of power to shared message</li></ul>Interference Cancellation<br /><ul><li>Dirty paper coding to cancel interference </li></ul>Support Interference Channel Transmission<br /><ul><li>Han & Kobayashi coding to cope with interference like channel</li></li></ul><li>Coding <br />Legitimate Transmitter<br />Legitimate Receiver<br />Codebook1<br />Joint <br />Decoder<br />Codebook2<br />Rate <br />Split<br />Codebook3<br />Codebook1<br />Joint <br />Decoder<br />DPC<br />Rate <br />Split<br />Codebook4<br />SIC<br />Cognitive Transmitter<br />Cognitive Receiver<br />
  • 69. Numerical Result<br />Performance of achievable scheme is compared to the outer bound <br />Interference gain<br /><ul><li>b = 0.5
  • 70. a = 2</li></ul>Power constraints<br /><ul><li>Power constraint for legitimate transmitter: 10
  • 71. Power constraint for cognitive transmitter: 10</li></li></ul><li>Numerical Result<br /> : Approaches Capacity Region of Fully Cognitive Radio<br /> : Good Inner and Outer Bound for Interference Channel<br />
  • 72. Contribution<br />Capacity of Partially Cognitive Radio<br /><ul><li>‘Good’ Outer Bound
  • 73. Tight in one extreme: Fully Cognitive Radio
  • 74. Gradual decrease with loss of cognitive information
  • 75. ‘Good’ Achievable Scheme
  • 76. Capacity Achieving in one extreme: Fully Cognitive Radio
  • 77. Best Known Capacity Achieving Coding : Interference Channel
  • 78. Gradual decrease with loss of cognitive information</li></li></ul><li>Future Work<br />Sum Capacity with Constant Gap<br /><ul><li>Interference Channel : 1 Bit Gap ( R. Etkin et. al. )
  • 79. Capacity with 1 Bit gap in Gaussian interference channel
  • 80. Fully Cognitive Channel : Capacity Achieving
  • 81. Capacity achieving in Gaussian interference channel with Fully Cognitive Radio</li></ul>Partially Cognitive Radio<br /><ul><li>Generalize the Sum Capacity Result
  • 82. Establish sum capacity with constant gap in two extremes
  • 83. Establish sum capacity with a gap corresponding to </li></li></ul><li>References<br />J.Mitola, “Cognitive Radio,” Ph.D. dissertation, Royal Institute of Technology (KTH),Stockholm, Sweden, 2000.<br />S. Haykin, Cognitive radio: brain-empowered wireless communications, IEEE J. Sel. Areas in Commun., vol. 23, pp. 201-220, Feb. 2005.<br />N. Devroye, P. Mitran and V. Tarokh, “Achievable Rates in Cognitive Rado Channels,” IEEE Trans. Inform. Theory, vol. 52, pp. 1813-1827, May 2006.<br />I. Maric, A. Goldsmith, G. Kramer, S. Shamai (Shitz), “On the Capacity of Interference Channel with a Partially-Cognitive Traansmitter,”IEEE Trans. Inform. Theory.<br />I. Maric, R. Yates, “The Strong Interference Channel with Common Information,” Allerton Conf. Communications, Monticello, Il, Sep. 2005.<br />I. Maric, R. Yates, G. Kramer, “The strong interference channel with unidirectional cooperation,” presented at the Information Theory and Applications (ITA) Inaugural Workshop, Feb. 2006.<br />W. Wu, S. Vishwanath and A. Arapostathis, “On the Capacity of Interference Channel with degraded Message Sets,”IEEE Trans. Inform. Theory.<br />T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,”IEEE Trans. Inform. Theory, vol. 27, pp. 49-60, Jan. 1981.<br />H. Sato, “Two-user communication channels,” IEEE Trans. Inform. Theory, vol. 23, pp. 295-304, May 1977.<br />A. B. Carleial, “Outer bounds on the capacity of interference channels,” IEEE Trans. Inform. Theory, vol. 29, pp. 602-606, Jul. 1983. vol. IT-24,<br />J. A. Thomas, “Feedback can at most double Gaussian multiple access channel capacity,” IEEE Trans. Inform. Theory, vol. 33, pp. 711-716, Sep. 1987.<br />H. Weingarten, Y. Steinberg and S. Shamai (Shitz), “The Capacity region of the Gaussian MIMO broadcast channel,” IEEE Trans. Inform. Theory, vol. 52, pp. 3936-3964, Sep. 2006.<br />M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. 29, pp. 439-441, May 1983.<br />T. M Cover and J.A. Thomas, Elements of information theory, ser. Wiley Series in Telecommunications. New York: John Wiley & Sons Inc., 1991, a Wiley-Interscience Publication.<br />
  • 84. Part IICapacity of Interweave Cognitive Radio<br />
  • 85. Motivation<br />Low Utilization <br /><ul><li>FCC Report
  • 86. Several unused 6MHz channel everywhere
  • 87. Possibility of opening the channel near future</li></ul>Interweave Cognitive Radio <br /><ul><li>Opportunistic Access
  • 88. Increase spectral efficiency
  • 89. Channel Selection & Power allocation
  • 90. Establish maximum increase in spectral efficiency</li></li></ul><li>Multiple Channel<br />sense<br />sense<br />sense<br />Multiple Channel in Interweave Cognitive Radio<br /><ul><li>Total legitimate channels : N
  • 91. Number of Simultaneous Sensing : L
  • 92. Selection of L channels to sense from N legitimate channels
  • 93. Transmission over available channels</li></ul>Example when N=4 and L=2<br />: Licensed user<br />: Cognitive user<br />
  • 94. Channel Environment<br />Dissimilarity between Channels <br /><ul><li>Noise Variances:
  • 95. Different noise variances
  • 96. Represents fading states
  • 97. Probability of Channel Being Available:
  • 98. Different probabilities
  • 99. Low indicates frequently used channel</li></ul>Graphical Representation of Channel, N=4<br />
  • 100. Question<br />Selection of Channel Sensing <br /><ul><li>Low Noise Variance
  • 101. High Probability of Being Available</li></ul> What is a good channel in the mixture?<br />Power Allocation <br /><ul><li>Dependent on Channel Sensing
  • 102. No power allocation on unavailable channel </li></ul> How do we allocate power?<br />Joint Optimization <br /><ul><li>Power Allocation and Channel Selection
  • 103. Maximize Average Capacity</li></ul> Limit of interweave cognitive radio & Building of efficient system<br /><ul><li>Probability of Channel Being Available:
  • 104. Different probabilities
  • 105. Low indicates frequently used channel</li></li></ul><li>Joint Optimization Problem<br />Optimization Parameter <br /><ul><li>Channel Selection:
  • 106. Power Allocation :</li></ul>Average Capacity Maximization <br />
  • 107. <ul><li>Power Allocation with given
  • 108. Find power allocation when channel selection is given
  • 109. Graphically represented by “modified water-filling”</li></ul>Optimal Power Allocation<br />Example when N=4, L=3<br />
  • 110. <ul><li>Exhaustive Search
  • 111. Exhaustive search of channels with the highest capacity
  • 112. Combinatorial times “modified water-filling”
  • 113. Require more efficient method to select channels</li></ul>Optimal Channel Selection<br />
  • 114. <ul><li>Objective
  • 115. Approximate the optimal channel selection and power allocation
  • 116. Reduce complexity
  • 117. Two Step Approximation
  • 118. Coarse Optimization
  • 119. Lowest water-level
  • 120. Fine Optimization
  • 121. Neglect poor channels
  • 122. Use lowest water-level</li></ul>Sub-Optimal Channel Selection<br />Coarse Optimization<br />Optimal?<br />Yes<br />Lowest water-level<br />Coarse Optimization<br />
  • 123. <ul><li>Coarse Optimization
  • 124. Find channels that minimize water-level of modified water-filling
  • 125. Iteratively find channels which lower the water-level
  • 126. maximum N-L iteration
  • 127. Optimal if water-level is lower than any unselected channel</li></ul>Coarse Optimization<br />Arbitrary L channels & modified water-filling<br />Compare area under water-level<br />L largest channels & modified water-filling<br />Terminate if no larger area<br />Example when N=5, L=2<br />
  • 128. <ul><li>Intuition
  • 129. Existence of unselected channel with low noise variance
  • 130. Channel selection can be improved
  • 131. Existence of channel with very high noise variance
  • 132. Useless ( Does not increase capacity )
  • 133. Method
  • 134. Exclude useless channels
  • 135. Relax integer condition on channel selection
  • 136. Find convex function</li></ul>Fine Optimization<br />Coarse Optimization<br />Useless Channel<br />Fine Optimization Needed<br />
  • 137. Fine Optimization<br />Optimization Problem <br /><ul><li>Find channel selection & power allocation to maximize capacity
  • 138. : Minimum water level from coarse optimization
  • 139. k : Constant which makes the objective function convex</li></li></ul><li>2010년 4월 18일<br />Fine Optimization<br />Solution<br /><ul><li>Solution from KKT condition</li></ul>Optimality <br /><ul><li>Performs the same with the exhaustive search</li></ul>Complexity <br /><ul><li>Order N iteration</li></li></ul><li>Numerical Result<br />Performance Comparison<br /><ul><li>Exhaustive Search
  • 140. Coarse Optimization
  • 141. Fine Optimization</li></ul>Parameter <br /><ul><li>N=8
  • 142. L=4</li></li></ul><li><ul><li>Coarse Optimization
  • 143. Optimal in low SNR
  • 144. Fine Optimization
  • 145. Same performance with exhaustive search in all SNR region</li></ul>Numerical Result<br />
  • 146. Contribution<br />Average Capacity of Interweave Cognitive Radio<br /><ul><li>Joint Optimization to maximize average capacity
  • 147. Channel Selection
  • 148. Power Allocation
  • 149. Limits of Interweave Cognitive Radio is verified</li></ul>Practical Maximum Capacity Achieving System <br /><ul><li>Computationally practical algorithm
  • 150. Coarse Optimization
  • 151. Fine Optimization</li></ul> Practical interweave cognitive radio system is built<br /> Contribution to Spectral Efficiency<br />
  • 152. Future Work<br />Interweave Cognitive Radio with Learning<br /><ul><li>Exact Knowledge on Channel Availability Probability
  • 153. In practice, Cognitive may not know the exact probability
  • 154. Understanding of the probability can be enhanced by observation</li></ul>Exploitation & Exploration <br /><ul><li>Exploitation
  • 155. With the sensing result, cognitive radio makes a transmission
  • 156. Exploration
  • 157. With the sensing result, cognitive radio updates the probability of channel being available</li></li></ul><li>Joint Power Allocation & Channel Selection with Learning<br />sense<br />sense<br />sense<br />Interweave Cognitive Radio with Learning ( L. Lei et. al. )<br /><ul><li>Channel Selection with fixed power allocation
  • 158. Multi arm bandit solution</li></ul>: Licensed user<br />: Cognitive user<br />Frequency Time Power Allocation & Channel selection <br />
  • 159. References<br />S. Srivasa and S. Jafar, “The Throughput Potential of Cognitive Radio: A Theoretical Perspective,” Asilomar Conf. on Signals, Systems, and Computers, Asilomar, CA, Oct. 2006.<br />N. Devroye, P. Mitran, and V. Tarokh, “Achievable Rates in Cognitive Rado Channels,” IEEE Trans. Inform. Theory, vol. 52, pp. 1813-1827, May 2006.<br />W. Wu, S. Vishwanath, and A. Arapostathis, “Capacity of a Class of Cognitive Radio Channels: Interference Channels With Degraded Message Sets,” IEEE Trans. Inform. Theory, vol. 53, pp. 4391-4399, Nov. 2007.<br />W. Wang, T. Peng and W. Wang, “Optimal Power Control under Interference Temperature Constraints in Cognitive Radio Network,” IEEE Wireless Comm. & Networking Conf., Hong Kong, Mar. 2007.<br />A. Goldsmith and P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inform. Theory, vol. 43, pp. 1986-1992, Nov. 1997.<br />Y. Song, Y. Fang, and Y. Zhang, “Stochastic Channel Selection in Cognitive Radio Networks,” IEEE Global Communications Conf., Washington, DC, Nov. 2007.<br />X. Yang, Z. Yang, and D. Liao, “Adaptive Spectrum Selection for Cognitive Radio Networks,” International Conf. on Computer Science and Software Engineering, Wuhan, China, Dec. 2008.<br />D. Huang, C. Miao, C. Leung and Z. Shen, “Resource Allocation of MU-OFDM Based Cognitive Radio Systems Under Partial Channel State Information,” http://arxiv.org/abs/0808.0549<br />G. Chung, S. Vishwanath, and C. S. Hwang, “On the Fundamental Limits of Interweaved Cognitive Radios,” http://arxiv.org/abs/0910.1639<br />
  • 160. Part IIISensing of Interweave Cognitive Radio<br />
  • 161. Motivation<br />Sensing Accuracy<br /><ul><li>Accurate Cognitive Information
  • 162. Guarantee Legitimate User’s Reliable Transmission</li></ul>Cooperative Sensing <br /><ul><li>Firm Requirement on Interference to Legitimate User
  • 163. Cooperative sensing increases sensing ability</li></ul>Spectral Efficiency <br /><ul><li>Trade offs between spectral efficiency and detection error
  • 164. Design sensing system to maximize spectral efficiency
  • 165. Keep detection error in tolerable level</li></li></ul><li>Cooperative Sensing<br />Legitimate Radio<br />Sensing Network <br /><ul><li>Energy Detection
  • 166. Each sensor detects energy level
  • 167. Underlay Network
  • 168. Unaware of existence of legitimate user
  • 169. No dedicated channel exists
  • 170. Limited bit and distance</li></ul>Sensing Network<br />
  • 171. Cooperative Sensing<br />Energy Detector<br />Decision<br />Parameter and Variables <br /><ul><li>Detection Time at each Detector
  • 172. Mapping to </li></ul>Legitimate<br />Transmitter<br />Energy Detector<br />& Mapping<br />Energy Detector<br />& Mapping<br />
  • 173. Previous Work<br />Distributed Computation ( Tsisikliset. al.) <br /><ul><li>General Model on Distributed Computation
  • 174. Solution to Sensing Error</li></ul>Sensing Algorithm <br /><ul><li>Probability of Detection
  • 175. Probability of False Alarm</li></li></ul><li>Future Work<br />Build Cooperative Sensing System <br /><ul><li>Spectrally Efficient System
  • 176. Limit on Detection Error</li></ul>Expected Contribution<br /><ul><li>Combined work of Sensing Network and Spectrally Efficient System
  • 177. Spectrally Efficient System with Constraint on Cognitive Radio</li></li></ul><li>Summary <br />Capacity Analysis of Cognitive Radio<br /><ul><li>Capacity of Overlay Cognitive Radio
  • 178. ‘Good’ Outer Bound
  • 179. ‘Good’ Inner Bound
  • 180. Capacity with a Reasonable Gap
  • 181. Maximum Average Capacity of Interweave Cognitive Radio</li></ul>System Design of Cognitive Radio<br /><ul><li>Build a Spectrally Efficient Cognitive Radio System
  • 182. Coding Strategy for Overlay Cognitive Radio
  • 183. Resource Allocation Algorithm for Interweave Cognitive Radio</li></ul> - Exact Knowledge of probability<br /> - With Learning<br /><ul><li>Sensing Strategy for Interweave Cognitive Radio </li></li></ul><li>Appendix (1)<br />Proof of outer bound (IFC-DMS)<br />(a)from conditional markov chain <br />(b)from identifying <br />
  • 184. Appendix (2)<br />(a)from identifying <br />
  • 185. Appendix (3)<br />(a)from markov chain <br />(c)from identifying <br />(b)from conditional markov chain <br />
  • 186. Appendix (4)<br />(a)from identifying <br />
  • 187. Appendix (5)<br />(a)from markov chain <br />(c)from identifying <br />(b)from conditional markov chain <br />
  • 188. Appendix (6)<br />Proof of outer bound (Gaussian)<br />Lemma: Let be arbitrarily distributed zero-mean random variables with covariance matrix , where are independent of each other. Let be the zero-mean Gaussian distributed random variables with the same covariance matrix. Then,<br /> Lemma: Let be arbitrarily distributed zero-mean random variables , and be the Gaussian distributed random variables with the same covariance matrix. Let be any subset of {1,2,…,k} and be its complement. Then,<br /> With help of EPI and above Lemma, outer bound can be proven<br />

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