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On AVCs WITH Quadratic Constraints. Farzin Haddadpour Joint work with Madhi Jafari Siavoshani , Mayank Bakshi and Sidharth Jaggi. Sharif University of Technology, Iran ISSL, EE Department. Institute of Network Coding The Chinese University of Hong Kong. 2013 ISIT July 7, 2013 . - PowerPoint PPT Presentation
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ON AVCS WITH QUADRATIC CONSTRAINTS
Farzin HaddadpourJoint work with
Madhi Jafari Siavoshani, Mayank Bakshi and Sidharth Jaggi
Sharif University of Technology, IranISSL, EE Department
2013 ISITJuly 7, 2013
Institute of Network CodingThe Chinese University of Hong Kong
Outline
1/18
•Introduction•System Model•Relation with Prior Works
•Main Result•Proof Steps•Conclusion
Introduction
2/18
Goal: decode message
Goal: interrupt Alice’s information of their movement
Goal: transmitreliably
How can I interrupt this transmission?
Alice
Willie
Bob
System Model
3/18
Enc Deci j( )nx i
ns nV
Power Constraints: 2|| ( ) ||nx i nP2|| ||ns n
nV : i.i.d. Gaussian Vector 2(0, )
ny
Y X s V
Prior Works
4/18
[Hughes and Narayan 1988]
Enc Dec
Jammer
i
Shared common randomness
i
Message Aware Jammer
nV
Capacity Rate:2
1 log(1 )2
P
Prior Works
5/18
[Csizar and Narayan 1991]
Capacity Rate:2
1 log(1 )2
0
P
if P
otherwise
i iEnc Dec
JammernV
ns
Our Model
6/18
Enc Dec
JammernV
ns
Enc Dec
Jammer
i
Shared common randomness
i
Message Aware Jammer
nV
i i
Our Model
7/18
Enc Dec
Jammer
Private randomization
Message- aware Jamming
nV
( , )nx i t
t0n
( )ns i
•Stochastic encoding•Public code•Message-aware jamming•Oblivious adversary
ii
Main Result
8/18
Enc Dec
Jammer
i i
Private randomization
Message- aware Jamming
nV
( , )nx i t
t0n
( )ns i
2
1 log(1 )2
0
P
if
otherwise
P Theorem(Capacity Rate):
Achievability Proof
9/18
•Codebook : 11
2 ... 0ne
2
.
nRe
..
(1,1)x 0(1, )nx e
( ,1)nRx e 0( , )n nRx e e
( , )x i t ( , '')x i t
( , ')x j tError
No Error
•Intuition : Because of our error probability we take average over colored row otherwise Csizar’s approach which has averaging over whole codewords
•Note: Decoder uses ML decoding
( )0i
y
if for
2 2|| || || || ,i jy x y x
:1i i M
i j
if no such exists
Achievability Proof
10/18
•Based on this Criteria error probability is:2 2( , ) [|| ( , ) ( , ') || || ||T Ve s i p p x i T s V x j t s V
for some andi j ']t
[ ( , ) , ( , ') ( , ),T Vp p x i T s V x j t nP x i T s V for some andi j ']t
•Lemma1: fix vector then for every and
*{(,)} Xit
( , )X i t uniformly distributed over (0, )n nP
1 ,nRi e 01 nt e
(0, )ns n 1 00
n 2
1 log(1 )2
PR
for large if
1* 0 1[ ( , ) ] exp( ( log 2 10)exp(( ) ))np e s i Ke K n
Achievability Proof(Lemma1)•Proof of Lemma 1 :
Lemma A1 [Csizar and Narayan 1991] : Let be arbitrary r.v.’s and be arbitrary function with then the condition a,s, implies
1. Using Lemma A1 and taking
we have
11/18
1,..., LX X
1( ,..., )i if X X 0 1, 1,...,if i L 1 1 1[ ( ,..., ) | ,..., ]i i iE f X X X X a 1,...,i L
1
1[ ( ( ,1),..., ( , )) ] exp( ( log 2 ))L
tt
p f X i X i t L aL
( ( ,1),..., ( , )) [ ( , ) , ( , ') ( , ),t Vf X i X i t p x i T s V x j t nP x i T s V
for some andi j 1'] ]nt Ke * [ [ ( , ) , ( , ') ( , ),T Vp p p x i T s V x j t nP x i T s V
0
*0
1
1[ [ [ ( , ) , ( , ') ( , ),ne
Vnt
p p x i t s V x j t nP x i t s Ve
for some andi j ']t
for some andi j 1'] ]nt Ke 0
1*
01
1[ [ ( ( ,1),..., ( , ))] ]ne
ntn
t
p f X i X i t Kee
Achievability Proof(Lemma1)2. So it remains to bound
Where (a) follows by .
12/18
1 1 1[ ( ,..., ) | ,..., ]i i iE f X X X X
*
( , '):
[ [ { ( , '), ( , ) ( , ), }]]Vj t j i
p p X j t X i t s V P X i t s V
*
)
(
(
)
2
1
[ ( , ), ]a
Vp p X i t s V
* 2( , '):
(2)
[ { ( , '), ( , ) ( , ), }, ( , ), ]]Vj t j i
p p X j t X i t s V P X i t s V X i t s V
[ ] [ [ ]] [ ] [ ]p p p p
Achievability Proof(Lemma1)
13/18
Then terms (1) and (2) can be upper bounded using this Lemma.
U
u
1( )2 2[| , | ] 2(1 )n
p U u
1 1
2 n
Lemma [Csizar and Narayan 1991]: u is a fix vector and U is distributed uniformly over sphere and for have
Achievability Proof(Lemma2)
14/18
Lemma 2(Quantizing Adversarial Vector): for a fixed vector , sufficient small and for every there exists a fixed codebook with rate which also does well for every .
Proof of Lemma 2: choosing where is a random vector over unit sphere and , then we can show that
2
1 log(1 )2
PR
' ( , )ns s
's s u u[ , ] 1( ', ) exp( )e s i K n
ssX V
1 00 { ( , )}X i ts
Achievability Proof(Lemma3)
15/18
Lemma 3(Codebook Existence): For every and enough large , there exist a fixed codebook with rate such that forevery vector , and every transmitted message :
Proof of Lemma 3: It’s enough to show that
But using Lemma 2 we don’t need to check for every but only for that covers , therefore we can write
1 00 n{ ( , )}X i t
2
1 log(1 )2
PR
s i
1( , ) ne s i Ke
* * 1lim inf [ , ( , ) exp( )] 0n p s i e i s K n
s net n (0, )n n
* * * *1 1[ , ( , ) exp( )] 1 [ , ( , ) exp( )]n np s i e i s K n p s i e i s K n
* *
( )
11
1 [ ( , ) exp( )]nR
n
ea
si
p e i s K n
Union bound
Achievability Proof(Lemma3)Consider this figure for upper bounding the Cardinality of
16/18
net n
* * 1[ , ( , ) exp( )]np s i e i s K n
0 121 ( ) exp( ) exp( 'exp( ( )))nn nR K n
Conclusion
17/18
Such as Discrete Scenarios Using
Stochastic Encoder won’t Improve Capacity Region
THANKS FOR CONSIDERATIONAny
Questions?
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