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  1. 1. Fundamental Limitsof Cognitive Radios<br />Goochul Chung<br />
  2. 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. 3. Stringent Regulation on Interference to Legitimate Radio</li></ul>Interference<br /><ul><li>Interference to Legitimate Radio
  4. 4. Active Legitimate Radio
  5. 5. Transmission over Legitimate Channel (Without Cooperation)</li></ul>Legitimate Radio<br />Secondary Network<br />
  6. 6.
  7. 7. Motivation for Cognitive Radio<br />Scarcity of Channel<br /><ul><li>Almost all wireless channels have licensed user with utmost privilege
  8. 8. Constraint for new wireless applications</li></ul>Cognitive Radio<br /><ul><li>Possess Cognitive Information about Legitimate User
  9. 9. Interference Avoidance
  10. 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. 11. Interference Level ( Channel Gain )
  12. 12. Simultaneous Transmission with Limited Interference</li></ul>Interweave Cognitive Radio<br /><ul><li>Cognitive Information
  13. 13. Channel Availability
  14. 14. No Simultaneous Transmission </li></ul>Overlay Cognitive Radio<br /><ul><li>Cognitive Information
  15. 15. Messages of Legitimate User
  16. 16. Simultaneous Transmission </li></li></ul><li>Underlay Cognitive Radio<br />Underlay Cognitive Radio<br /><ul><li>Ultra Wide Band
  17. 17. Spread its message over ultra wide band ( 3 – 10 GHz )
  18. 18. Small Power
  19. 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. 20. Interweave Cognitive Radio<br />Interweave Cognitive Radio<br /><ul><li>No Simultaneous Transmission
  21. 21. Sense if there is legitimate radio
  22. 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. 23. Overlay Cognitive Radio<br /><ul><li>Knowledge of Legitimate Users Message
  24. 24. Obtain legitimate user’s message sets
  25. 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. 26. Question<br />Cognitive Information Acquisition <br /><ul><li> Selective Information Acquisition
  27. 27. Can information be efficiently collected to increase spectral efficiency? ( Channel Selection )
  28. 28. Sensing
  29. 29. How can sensing increase spectral efficiency with sensing accuracy?</li></ul>Utilization of Cognitive Information<br /><ul><li> Resource Allocation
  30. 30. How can the resource (power) be allocated to increase average capacity?
  31. 31. Coding
  32. 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. 33. Limits of Cognitive Radio
  34. 34. Overlay Cognitive Radio
  35. 35. Interweave Cognitive Radio
  36. 36. Underlay Cognitive Radio</li></ul>System Design of Cognitive Radio<br /><ul><li>Build a Spectrally Efficient Cognitive Radio System
  37. 37. Coding Strategy for Overlay Cognitive Radio
  38. 38. Resource Allocation Algorithm for Interweave Cognitive Radio
  39. 39. Sensing Strategy for Interweave Cognitive Radio </li></li></ul><li>Part ICapacity of Overlay Cognitive Radio<br />
  40. 40. Previous Works<br />Fully Cognitive Radio<br /><ul><li> Cognitive of the entire message set of legitimate user
  41. 41. Capacity of strong interference case (I. Maricet. al.)
  42. 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. 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. 44. : message set known to cognitive radio
  45. 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. 46. Motivation<br />Limited Cognitive Information<br /><ul><li>Practicality
  47. 47. Knowledge of full non-causal message set is not guaranteed
  48. 48. Generalization
  49. 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. 50. Knowledge of limits of the channel
  51. 51. Coding Design
  52. 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. 53. : Fully Cognitive Radio Channel
  54. 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. 55. Unidirectional cooperation from the cognitive transmitter
  56. 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. 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. 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. 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. 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. 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. 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. 63. Outer Bound (Gaussian Channel)<br />Outer bound of capacity region<br /> : convex closure of set : Transmit powers <br />
  64. 64. Outer Bound (Gaussian Channel)<br />Encoding With Power Split<br /><ul><li>Constraint on : Transmit cooperation with power split
  65. 65. Same codeword for
  66. 66. Constraint on : Interference free for transmission
  67. 67. Interference cancellation at the transmitter </li></ul>Decoder1<br />Decoder2<br />Encoding With Power Split & Interference Cancellation<br />
  68. 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. 69. Numerical Result<br />Performance of achievable scheme is compared to the outer bound <br />Interference gain<br /><ul><li>b = 0.5
  70. 70. a = 2</li></ul>Power constraints<br /><ul><li>Power constraint for legitimate transmitter: 10
  71. 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. 72. Contribution<br />Capacity of Partially Cognitive Radio<br /><ul><li>‘Good’ Outer Bound
  73. 73. Tight in one extreme: Fully Cognitive Radio
  74. 74. Gradual decrease with loss of cognitive information
  75. 75. ‘Good’ Achievable Scheme
  76. 76. Capacity Achieving in one extreme: Fully Cognitive Radio
  77. 77. Best Known Capacity Achieving Coding : Interference Channel
  78. 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. 79. Capacity with 1 Bit gap in Gaussian interference channel
  80. 80. Fully Cognitive Channel : Capacity Achieving
  81. 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. 82. Establish sum capacity with constant gap in two extremes
  83. 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. 84. Part IICapacity of Interweave Cognitive Radio<br />
  85. 85. Motivation<br />Low Utilization <br /><ul><li>FCC Report
  86. 86. Several unused 6MHz channel everywhere
  87. 87. Possibility of opening the channel near future</li></ul>Interweave Cognitive Radio <br /><ul><li>Opportunistic Access
  88. 88. Increase spectral efficiency
  89. 89. Channel Selection & Power allocation
  90. 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. 91. Number of Simultaneous Sensing : L
  92. 92. Selection of L channels to sense from N legitimate channels
  93. 93. Transmission over available channels</li></ul>Example when N=4 and L=2<br />: Licensed user<br />: Cognitive user<br />
  94. 94. Channel Environment<br />Dissimilarity between Channels <br /><ul><li>Noise Variances:
  95. 95. Different noise variances
  96. 96. Represents fading states
  97. 97. Probability of Channel Being Available:
  98. 98. Different probabilities
  99. 99. Low indicates frequently used channel</li></ul>Graphical Representation of Channel, N=4<br />
  100. 100. Question<br />Selection of Channel Sensing <br /><ul><li>Low Noise Variance
  101. 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. 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. 103. Maximize Average Capacity</li></ul> Limit of interweave cognitive radio & Building of efficient system<br /><ul><li>Probability of Channel Being Available:
  104. 104. Different probabilities
  105. 105. Low indicates frequently used channel</li></li></ul><li>Joint Optimization Problem<br />Optimization Parameter <br /><ul><li>Channel Selection:
  106. 106. Power Allocation :</li></ul>Average Capacity Maximization <br />
  107. 107. <ul><li>Power Allocation with given
  108. 108. Find power allocation when channel selection is given
  109. 109. Graphically represented by “modified water-filling”</li></ul>Optimal Power Allocation<br />Example when N=4, L=3<br />
  110. 110. <ul><li>Exhaustive Search
  111. 111. Exhaustive search of channels with the highest capacity
  112. 112. Combinatorial times “modified water-filling”
  113. 113. Require more efficient method to select channels</li></ul>Optimal Channel Selection<br />
  114. 114. <ul><li>Objective
  115. 115. Approximate the optimal channel selection and power allocation
  116. 116. Reduce complexity
  117. 117. Two Step Approximation
  118. 118. Coarse Optimization
  119. 119. Lowest water-level
  120. 120. Fine Optimization
  121. 121. Neglect poor channels
  122. 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. 123. <ul><li>Coarse Optimization
  124. 124. Find channels that minimize water-level of modified water-filling
  125. 125. Iteratively find channels which lower the water-level
  126. 126. maximum N-L iteration
  127. 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. 128. <ul><li>Intuition
  129. 129. Existence of unselected channel with low noise variance
  130. 130. Channel selection can be improved
  131. 131. Existence of channel with very high noise variance
  132. 132. Useless ( Does not increase capacity )
  133. 133. Method
  134. 134. Exclude useless channels
  135. 135. Relax integer condition on channel selection
  136. 136. Find convex function</li></ul>Fine Optimization<br />Coarse Optimization<br />Useless Channel<br />Fine Optimization Needed<br />
  137. 137. Fine Optimization<br />Optimization Problem <br /><ul><li>Find channel selection & power allocation to maximize capacity
  138. 138. : Minimum water level from coarse optimization
  139. 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. 140. Coarse Optimization
  141. 141. Fine Optimization</li></ul>Parameter <br /><ul><li>N=8
  142. 142. L=4</li></li></ul><li><ul><li>Coarse Optimization
  143. 143. Optimal in low SNR
  144. 144. Fine Optimization
  145. 145. Same performance with exhaustive search in all SNR region</li></ul>Numerical Result<br />
  146. 146. Contribution<br />Average Capacity of Interweave Cognitive Radio<br /><ul><li>Joint Optimization to maximize average capacity
  147. 147. Channel Selection
  148. 148. Power Allocation
  149. 149. Limits of Interweave Cognitive Radio is verified</li></ul>Practical Maximum Capacity Achieving System <br /><ul><li>Computationally practical algorithm
  150. 150. Coarse Optimization
  151. 151. Fine Optimization</li></ul> Practical interweave cognitive radio system is built<br /> Contribution to Spectral Efficiency<br />
  152. 152. Future Work<br />Interweave Cognitive Radio with Learning<br /><ul><li>Exact Knowledge on Channel Availability Probability
  153. 153. In practice, Cognitive may not know the exact probability
  154. 154. Understanding of the probability can be enhanced by observation</li></ul>Exploitation & Exploration <br /><ul><li>Exploitation
  155. 155. With the sensing result, cognitive radio makes a transmission
  156. 156. Exploration
  157. 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. 158. Multi arm bandit solution</li></ul>: Licensed user<br />: Cognitive user<br />Frequency Time Power Allocation & Channel selection <br />
  159. 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. 160. Part IIISensing of Interweave Cognitive Radio<br />
  161. 161. Motivation<br />Sensing Accuracy<br /><ul><li>Accurate Cognitive Information
  162. 162. Guarantee Legitimate User’s Reliable Transmission</li></ul>Cooperative Sensing <br /><ul><li>Firm Requirement on Interference to Legitimate User
  163. 163. Cooperative sensing increases sensing ability</li></ul>Spectral Efficiency <br /><ul><li>Trade offs between spectral efficiency and detection error
  164. 164. Design sensing system to maximize spectral efficiency
  165. 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. 166. Each sensor detects energy level
  167. 167. Underlay Network
  168. 168. Unaware of existence of legitimate user
  169. 169. No dedicated channel exists
  170. 170. Limited bit and distance</li></ul>Sensing Network<br />
  171. 171. Cooperative Sensing<br />Energy Detector<br />Decision<br />Parameter and Variables <br /><ul><li>Detection Time at each Detector
  172. 172. Mapping to </li></ul>Legitimate<br />Transmitter<br />Energy Detector<br />& Mapping<br />Energy Detector<br />& Mapping<br />
  173. 173. Previous Work<br />Distributed Computation ( Tsisikliset. al.) <br /><ul><li>General Model on Distributed Computation
  174. 174. Solution to Sensing Error</li></ul>Sensing Algorithm <br /><ul><li>Probability of Detection
  175. 175. Probability of False Alarm</li></li></ul><li>Future Work<br />Build Cooperative Sensing System <br /><ul><li>Spectrally Efficient System
  176. 176. Limit on Detection Error</li></ul>Expected Contribution<br /><ul><li>Combined work of Sensing Network and Spectrally Efficient System
  177. 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. 178. ‘Good’ Outer Bound
  179. 179. ‘Good’ Inner Bound
  180. 180. Capacity with a Reasonable Gap
  181. 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. 182. Coding Strategy for Overlay Cognitive Radio
  183. 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. 184. Appendix (2)<br />(a)from identifying <br />
  185. 185. Appendix (3)<br />(a)from markov chain <br />(c)from identifying <br />(b)from conditional markov chain <br />
  186. 186. Appendix (4)<br />(a)from identifying <br />
  187. 187. Appendix (5)<br />(a)from markov chain <br />(c)from identifying <br />(b)from conditional markov chain <br />
  188. 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 />