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

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

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

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.

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.

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.

June 4, 2015 Cognitive Radio Network Architecture

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

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

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)

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?

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

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

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

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.

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.

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

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

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

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.

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

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

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

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

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

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

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.

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)

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

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

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

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

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.

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

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

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

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

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

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