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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
April 18, 2023
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
April 18, 2023
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.
April 18, 2023
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.
April 18, 2023
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
April 18, 2023
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.
April 18, 2023
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.
April 18, 2023
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
April 18, 2023
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
April 18, 2023
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
April 18, 2023
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.
April 18, 2023
Future Work
Show optimality of Achievable region for the remaining portion of the capacity region.
April 18, 2023
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
April 18, 2023
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
April 18, 2023
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
