june 4, 2015 on the capacity of a class of cognitive radios sriram sridharan in collaboration with...

April 18, 2023

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

April 18, 2023

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

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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

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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.

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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.

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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.

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Cognitive Radio Network Architecture

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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

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Cognitive Radio System Model

Licensed Transmitter : Message Transmits Power Constraint :

Cognitive Transmitter Message Transmits Power Constraint :

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Cognitive Radio System Model (Contd.)

System described by

Noise , are Gaussian Noise ~ N(0, 1)

Cognitive transmitter knows and (the codeword of licensed user)

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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?

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Capacity of Cognitive Channel The capacity of the cognitive channel is

[Viswanath et. al], [Wei Wu et.al.]

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Achievability Cognitive user allocates a portion of power ( Pc) 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

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MIMO Cognitive Radio Channel

Channel model similar to single antenna case

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MIMO Cognitive Radio System Model

MIMO cognitive radio (Channel Equations) Yp = Hp,p Xp + Hc,p Xc + Zp

Yc = Hp,c Xp + Hc,c Xc + Zc

np,t , np,r : Number of antennas for licensed user

nc,t, nc,r : Number of antennas for cognitive user

Gaussian Noise : Zp, Zc ~ N(0, I). Correlation between Zp and Zc arbitrary.

Channel gain matrices known at transmitter and receiver.

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MIMO Cognitive Radio System Model (Contd.)

Covariance matrices of codewords : p, c

Power constraints : Tr (p) · Pp

Tr (c) · Pc

Rate pair (Rp, Rc) is achievable if there exists

There exists decoders Dp, and Dc s.t. probability of decoding error is arbitrarily small.

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Achievable Region

Let be the set of rate pairs (Rp, Rc)

is achievable

G = [Hp,p Hc,p]

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Achievable Region (Contd.)

Similar to single antenna case

Hp,p

Hc,p

Hc,p

Hc,c

Xp

Xc,p

Xc,c

Pp

Pc

(1-) Pc

mp

mc

Costa Precoder

Zp

Zc

Costa Decoder

mpSingle UserDecoder

mc

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Remarks on Achievable Region

Optimization over covariance matrices (p, c,p, c,c)

Optimization over

Practical coding schemes

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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.

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Outer Bound (Transformation 1)

Licensed User : Licensed User :

Cognitive User : Cognitive User :

Power Constraint : Pp, Pc Power Constraint : Pp, Pc

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Outer Bound (Transformation 2)

Licensed User : Licensed User :

Cognitive User : Cognitive User :

Modified version of Ypn provided to cognitive receiver

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Licensed User : Licensed User :

Cognitive User : Cognitive User :

Outer Bound (Transformation 3)

We remove part of link from licensed transmitter to cognitive receiver

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Outer Bound (Transformation 4)

Licensed User : Licensed User :

Cognitive User : Cognitive User :

Allow transmitters to co-operate, Sum power constraint

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Outer Bound (Transformation 5)

Licensed User : Licensed User :

Cognitive User : Cognitive User :

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Outer Bound Region

Let be the convex hull of the set of rate pairs given

by

where , Then, is an outer bound

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Optimality of Achievable Region

Rate pair (Rp, Rc) lies on the boundary of capacity region If it maximizes Rp + Rc for some > 0

We show that our achievable region is – sum optimal for all ¸ 1

Let maximize Rp + Rc over the achievable region.

Then, is an element of for any > 0.

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Optimality of Achievable Region Optimization Problem 1

Rp, Rc, p, c,c, c,p such that

max Rp + Rc

We find the rate pair that maximizes Rp + Rc in achievable region

Let optimal value = M (bounded)

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Optimality of Achievable RegionLagrangian dual of Optimization Problem 1

Max min Rp + Rc - 1 (Tr(p) – Pp) - 2 (Tr (c,c

) + Tr(c,p

) – Pc)

Rp, Rc, p, c,c, c,p 1 > 0, 2 > 0

Let optimal value = U U ¸ M

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Optimality of Achievable RegionOptimization Problem 2

min max Rp + Rc

> 0

Let optimal value = N

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Optimality of Achievable RegionLagrangian Dual of Optimization Problem 2

Max min Rp + Rc - (Tr(p) + Tr(c,c) + Tr(c,p

) – Pp – P

c)

Rp, Rc, p, c,c, c,p > 0, > 0

Let optimal value = V V ¸ N

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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

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Challenges in Model

Assumption that mp is available non causally to cognitive transmitter

Possible only if cognitive transmitter is close to licensed transmitter.

Let Cpt, ct be capacity of link between licensed and cognitive transmitter

Let Cpt, pr be capacity of link between licensed transmitter and licensed

receiver

Cognitive transmitter acquires message mp faster than licensed receiver.

Channel gain matrices are known everywhere.

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Future Work

Show optimality of Achievable region for the remaining portion of the capacity region.

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Future Work (Contd.)

Assume no knowledge of mp at the cognitive transmitter

Cognitive transmitter transmits in the null space of Hc,p

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Achievable Region

Encoding Rule for Licensed User :

Generate Xpn(mp) according to the distribution

The covariance matrix p satisfies

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Achievable Region (Contd.)

Encoding Rule for Cognitive User

Stage 1 : Generate Xc,pn(mp) according to where

Stage 2 : Generate Xc,cn(mc) using Costa precoding

by treating Hp,c Xpn + Hc,c Xc,p

n as non causal interference.

Xc,cn is statistically independent of Xc,p

n, and

Xc,cn is distributed as where

Superposition : Xc

n = Xc,pn + Xc,c

n , where

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Achievable Region (Contd.)

Decoding Rule for Licensed Receiver

Receives Hp,p Xpn + Hc,p (Xc,p

n + Xc,cn) + Zp

n

Treats Hc,p Xc,cn + Zp

n as Gaussian noise.

Let G = [Hp,p Hc,p], where

Reliable decoding possible if

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Achievable Region (Contd.)

Decoding Rule for Cognitive Receiver

Cognitive decoder is Costa Decoder with knowledge of Ecn

Receives Ycn = Hp,c Xp

n + Hc,c (Xc,pn + Xc,c

n) + Zcn

Non causal interference Hp,c Xpn + Hc,c Xc,p

n cancelled by Costa precoder.

Reliable decoding possible if