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June 4, 2015 On the Capacity of a Class of Cognitive Radios Sriram Sridharan in collaboration with Dr. Sriram Vishwanath Wireless Networking and Communications.

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Presentation on theme: "June 4, 2015 On the Capacity of a Class of Cognitive Radios Sriram Sridharan in collaboration with Dr. Sriram Vishwanath Wireless Networking and Communications."— Presentation transcript:

1 June 4, 2015 On the Capacity of a Class of Cognitive Radios Sriram Sridharan in collaboration with Dr. Sriram Vishwanath Wireless Networking and Communications Group University of Texas at Austin

2 June 4, 2015 Inefficient Spectrum Utilization Spectrum occupancy averaged over 6 locations Spectrum is not efficiently utilized Dynamic Increase in utilization of limited spectrum for mobile services Effectiveness of traditional Spectrum policies strained Fig : “Cognitive Radio using Software … “- Dr. Jeffrey H. Reed

3 June 4, 2015 Dynamic Spectrum Access Networks (DSANs)  Proposed to solve spectrum inefficiency problems.  They provide high BW to mobile users via  Dynamic spectrum access techniques  Inefficiency in spectrum usage can be improved  through opportunistic access to existing licensed bands

4 June 4, 2015 Cognitive Radio  Terminology first coined by Joseph Mitola III and Gerald Q. Maguire, Jr.  Can be thought of as “fully reconfigurable wireless black box”  that can adapt to network and user demands.  Is a paradigm for Dynamic Spectrum Access Networks.

5 June 4, 2015 Original Idea of Cognitive Radio  Provide capability to use or share spectrum opportunistically.  Cognitive radio technology enabled users to  determine best portion of spectrum available for operation  detect the presence of licensed users in licensed band (spectrum sensing)  select best available channel (spectrum management)  co-ordinate access to channel with other users (spectrum sharing)  vacate channel when licensed user is detected.

6 June 4, 2015 Can we do better ?  We look at a model where cognitive radios do not vacate spectrum when licensed user arrives.  Can we still control interference (minimize rate loss)?  Knowledge of channel gain matrices.  Knowledge about licensed user’s transmissions.

7 June 4, 2015 Cognitive Radio Network Architecture

8 June 4, 2015 Fundamental Limits of Operation of Cognitive Radio Network  This model studied by [Tarokh et. al.], [Kramer et. al.], [Jovicic, Viswanath], [Wei Wu et. al], This is an Interference Channel with degraded message sets

9 June 4, 2015 Cognitive Radio System Model  Licensed Transmitter : Message Transmits Power Constraint :  Cognitive Transmitter Message Transmits Power Constraint :

10 June 4, 2015 Cognitive Radio System Model (Contd.)  System described by  Noise, are Gaussian Noise ~ N(0, 1)  Cognitive transmitter knows and (the codeword of licensed user)

11 June 4, 2015 Capacity of Cognitive Radio  Largest rate achieved by Cognitive User so that  No rate loss is caused to the licensed user  The licensed user can use a single user decoder  What is the rate tradeoff between the two users? (or) What is the capacity region of the cognitive user channel?

12 June 4, 2015 Capacity of Cognitive Channel  The capacity of the cognitive channel is [Viswanath et. al], [Wei Wu et.al.]

13 June 4, 2015  Achievability  Cognitive user allocates a portion of power (  P c ) to help the licensed user.  Cognitive transmitter uses Costa’s precoding scheme to nullify known interference  Converse  The capacity of Interference channel with degraded message sets is found (when a < 1). Proof Outline

14 June 4, 2015 MIMO Cognitive Radio Channel  Channel model similar to single antenna case

15 June 4, 2015 MIMO Cognitive Radio System Model  MIMO cognitive radio (Channel Equations) Y p = H p,p X p + H c,p X c + Z p Y c = H p,c X p + H c,c X c + Z c  n p,t, n p,r : Number of antennas for licensed user  n c,t, n c,r : Number of antennas for cognitive user  Gaussian Noise : Z p, Z c ~ N(0, I). Correlation between Z p and Z c arbitrary.  Channel gain matrices known at transmitter and receiver.

16 June 4, 2015 MIMO Cognitive Radio System Model (Contd.)  Covariance matrices of codewords :  p,  c  Power constraints : Tr (  p ) · P p Tr (  c ) · P c  Rate pair (R p, R c ) is achievable if there exists   There exists decoders D p, and D c s.t. probability of decoding error is arbitrarily small.

17 June 4, 2015 Achievable Region  Let be the set of rate pairs (R p, R c ) is achievable G = [Hp,p Hc,p]

18 June 4, 2015 Achievable Region (Contd.)  Similar to single antenna case Hp,p Hc,p Hc,c Xp Xc,p Xc,c Pp  Pc (1-  ) Pc mpmp mcmc Costa Precoder Zp Zc Costa Decoder mpmp Single User Decoder mcmc

19 June 4, 2015 Remarks on Achievable Region  Optimization over covariance matrices (  p,  c,p,  c,c )  Optimization over   Practical coding schemes

20 June 4, 2015 Outer Bound  Obtained by a series of channel transformations  Each transformation gives an outer bound.  Finally, we arrive at degraded broadcast channel  Its capacity region is the outer bound.

21 June 4, 2015 Outer Bound (Transformation 1) Licensed User : Cognitive User : Power Constraint : P p, P c Power Constraint : P p,  P c

22 June 4, 2015 Outer Bound (Transformation 2) Licensed User : Cognitive User : Modified version of Y p n provided to cognitive receiver

23 June 4, 2015 Licensed User : Cognitive User : Outer Bound (Transformation 3) We remove part of link from licensed transmitter to cognitive receiver

24 June 4, 2015 Outer Bound (Transformation 4) Licensed User : Cognitive User : Allow transmitters to co-operate, Sum power constraint

25 June 4, 2015 Outer Bound (Transformation 5) Licensed User : Cognitive User :

26 June 4, 2015 Outer Bound Region  Let be the convex hull of the set of rate pairs given by  where, Then, is an outer bound

27 June 4, 2015 Optimality of Achievable Region  Rate pair (R p, R c ) lies on the boundary of capacity region  If it maximizes  R p + R c for some  > 0  We show that our achievable region is  – sum optimal for all  ¸ 1  Let maximize  R p + R c over the achievable region.  Then, is an element of for any  > 0.

28 June 4, 2015 Optimality of Achievable Region Optimization Problem 1 R p, R c,  p,  c,c,  c,p such that max  R p + R c We find the rate pair that maximizes  R p + R c in achievable region Let optimal value = M (bounded)

29 June 4, 2015 Optimality of Achievable Region Lagrangian dual of Optimization Problem 1 Max min  R p + R c - 1 (Tr(  p ) – P p ) - 2 (Tr (  c,c ) + Tr(  c,p ) – P c ) R p, R c,  p,  c,c,  c,p 1 > 0, 2 > 0 Let optimal value = U U ¸ M

30 June 4, 2015 Optimality of Achievable Region Optimization Problem 2 min max  R p + R c  > 0 Let optimal value = N

31 June 4, 2015 Optimality of Achievable Region Lagrangian Dual of Optimization Problem 2 Max min  R p + R c - (Tr(  p ) + Tr(  c,c ) + Tr(  c,p ) – P p –  P c ) R p, R c,  p,  c,c,  c,p  > 0, > 0 Let optimal value = V V ¸ N

32 June 4, 2015 Optimality of Achievable Region  U = M  Power constraints are satisfied in Dual problem  V = N  Power constraint is satisfied in Dual problem  U = V  For every,  > 0, we have 1 =, 2 =   and vice versa  Hence, Achievable Region is  – sum optimal for all  ¸ 1

33 June 4, 2015 Challenges in Model  Assumption that m p is available non causally to cognitive transmitter  Possible only if cognitive transmitter is close to licensed transmitter.  Let C p t, c t be capacity of link between licensed and cognitive transmitter  Let C p t, p r be capacity of link between licensed transmitter and licensed receiver  Cognitive transmitter acquires message m p faster than licensed receiver.  Channel gain matrices are known everywhere.

34 June 4, 2015 Future Work  Show optimality of Achievable region for the remaining portion of the capacity region.

35 June 4, 2015 Future Work (Contd.)  Assume no knowledge of m p at the cognitive transmitter  Cognitive transmitter transmits in the null space of H c,p

36 June 4, 2015 Achievable Region  Encoding Rule for Licensed User :  Generate X p n (m p ) according to the distribution  The covariance matrix  p satisfies

37 June 4, 2015 Achievable Region (Contd.)  Encoding Rule for Cognitive User  Stage 1 : Generate X c,p n (m p ) according to where  Stage 2 : Generate X c,c n (m c ) using Costa precoding by treating H p,c X p n + H c,c X c,p n as non causal interference. X c,c n is statistically independent of X c,p n, and X c,c n is distributed as where  Superposition : X c n = X c,p n + X c,c n, where

38 June 4, 2015 Achievable Region (Contd.)  Decoding Rule for Licensed Receiver  Receives H p,p X p n + H c,p (X c,p n + X c,c n ) + Z p n  Treats H c,p X c,c n + Z p n as Gaussian noise.  Let G = [H p,p H c,p ], where  Reliable decoding possible if

39 June 4, 2015 Achievable Region (Contd.)  Decoding Rule for Cognitive Receiver  Cognitive decoder is Costa Decoder with knowledge of E c n  Receives Y c n = H p,c X p n + H c,c (X c,p n + X c,c n ) + Z c n  Non causal interference H p,c X p n + H c,c X c,p n cancelled by Costa precoder.  Reliable decoding possible if


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