Compute-and-Forward: Practical
Issues and Network Coding
Implementation
Koralia N. Pappi
Collaborating researcher, University of Patras, Greece,
E-mail: [email protected]
DISCO: Distributed Communication Systems
Chania, Crete, 15 September 2013
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Outline
1
Compute-and-Forward
Introduction
System Model
2
Channel Estimation Errors
Computation Rate Region with CSI error
Noisy Channel Estimates - Closed-form expressions
Numerical and Simulation Results
3
Physical Layer Network Coding
The optimal network coding vector
Reduction of candidate vectors
Offline Construction of Look-up Tables
4
Conclusions
DISCO: Distributed Communication Systems
15 September 2013
2 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Compute-and-Forward
Compute-and-Forward (C&F) Relaying: Interference
exploitation, high transmission rates.
C&F relay: Decoding of integer equations of the
transmitted messages.
C&F destination: Having enough independent
equations, it decodes the transmitted messages.
Tools:
Channel Estimation
Lattice Coding
PHY-Layer Network Coding (Channel State
Information is usually required)
DISCO: Distributed Communication Systems
15 September 2013
3 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Compute-and-Forward
Compute-and-Forward (C&F) Relaying: Interference
exploitation, high transmission rates.
C&F relay: Decoding of integer equations of the
transmitted messages.
C&F destination: Having enough independent
equations, it decodes the transmitted messages.
Tools:
Channel Estimation
Lattice Coding
PHY-Layer Network Coding (Channel State
Information is usually required)
DISCO: Distributed Communication Systems
15 September 2013
3 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Compute-and-Forward
Compute-and-Forward (C&F) Relaying: Interference
exploitation, high transmission rates.
C&F relay: Decoding of integer equations of the
transmitted messages.
C&F destination: Having enough independent
equations, it decodes the transmitted messages.
Tools:
Channel Estimation
Lattice Coding
PHY-Layer Network Coding (Channel State
Information is usually required)
DISCO: Distributed Communication Systems
15 September 2013
3 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Compute-and-Forward
Compute-and-Forward (C&F) Relaying: Interference
exploitation, high transmission rates.
C&F relay: Decoding of integer equations of the
transmitted messages.
C&F destination: Having enough independent
equations, it decodes the transmitted messages.
Tools:
Channel Estimation
Lattice Coding
PHY-Layer Network Coding (Channel State
Information is usually required)
DISCO: Distributed Communication Systems
15 September 2013
3 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
C&F Network
Each relay observes a noisy
linear combination of the
transmitted signals:
ym =
L
P
hm,l xl + zm .
l=1
The relay chooses a vector
am = [am,1 , am,2 , . . . , am,L ]T ,
am ∈ ZL , and decodes the
equation:
um = [
L
P
am,l xl ]modΛ.
l=1
DISCO: Distributed Communication Systems
15 September 2013
4 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
C&F Network
Each relay observes a noisy
linear combination of the
transmitted signals:
ym =
L
P
hm,l xl + zm .
l=1
The relay chooses a vector
am = [am,1 , am,2 , . . . , am,L ]T ,
am ∈ ZL , and decodes the
equation:
um = [
L
P
am,l xl ]modΛ.
l=1
DISCO: Distributed Communication Systems
15 September 2013
4 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Computation Rate Region
The relay multiplies the received signal by a parameter
αm .
The computation rate region achieved by the relay is
1
R(hm , am , αm ) = log+
2
2
P
2
αm + P kαm hm − am k2
,
The best choice for αm is the MMSE coefficient
βm = arg max [R (hm , am , αm )] =
αm
DISCO: Distributed Communication Systems
P hTm am
.
1 + P khm k2
15 September 2013
5 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Computation Rate Region
The relay multiplies the received signal by a parameter
αm .
The computation rate region achieved by the relay is
1
R(hm , am , αm ) = log+
2
2
P
2
αm + P kαm hm − am k2
,
The best choice for αm is the MMSE coefficient
βm = arg max [R (hm , am , αm )] =
αm
DISCO: Distributed Communication Systems
P hTm am
.
1 + P khm k2
15 September 2013
5 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Computation Rate Region
The relay multiplies the received signal by a parameter
αm .
The computation rate region achieved by the relay is
1
R(hm , am , αm ) = log+
2
2
P
2
αm + P kαm hm − am k2
,
The best choice for αm is the MMSE coefficient
βm = arg max [R (hm , am , αm )] =
αm
DISCO: Distributed Communication Systems
P hTm am
.
1 + P khm k2
15 September 2013
5 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Channel Estimation Errors
K. N. Pappi, G. K. Karagiannidis, and R. Schober, “How sensitive is compute and
forward to channel estimation errors?”, in Proc. IEEE International Symposium
on Information Theory (ISIT), Istanbul, Turkey, 7-12 July 2013.
If ĥm is the estimate of hm , the relay computes β̂m as:
β̂m =
P ĥTm am
1 + P kĥm k2
,
The MMSE coefficient error is defined as:
ε = β̂m − βm .
DISCO: Distributed Communication Systems
15 September 2013
6 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Channel Estimation Errors
K. N. Pappi, G. K. Karagiannidis, and R. Schober, “How sensitive is compute and
forward to channel estimation errors?”, in Proc. IEEE International Symposium
on Information Theory (ISIT), Istanbul, Turkey, 7-12 July 2013.
If ĥm is the estimate of hm , the relay computes β̂m as:
β̂m =
P ĥTm am
1 + P kĥm k2
,
The MMSE coefficient error is defined as:
ε = β̂m − βm .
DISCO: Distributed Communication Systems
15 September 2013
6 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Channel Estimation Errors
K. N. Pappi, G. K. Karagiannidis, and R. Schober, “How sensitive is compute and
forward to channel estimation errors?”, in Proc. IEEE International Symposium
on Information Theory (ISIT), Istanbul, Turkey, 7-12 July 2013.
If ĥm is the estimate of hm , the relay computes β̂m as:
β̂m =
P ĥTm am
1 + P kĥm k2
,
The MMSE coefficient error is defined as:
ε = β̂m − βm .
DISCO: Distributed Communication Systems
15 September 2013
6 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Computation Rate with Channel Estimation Errors
Theorem 1
2
If f (βm ) = βm
+ P kβm hm − am k2 then the computation
rate region of a C&F relay is
!
P
1
+
R(hm , am , ε) = log2
2
f (β̂m )
1
P
+
= log2
.
2
f (βm ) + ε2 (1 + P khm k2 )
DISCO: Distributed Communication Systems
15 September 2013
7 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Noisy Channel Estimation - High SNR Approximation
We consider noisy estimates of the form
ĥm = hm + nm ,
where nm is the channel estimation error vector of
length L, assumed to be Gaussian distributed with
N (0, σ 2 IL×L ).
For high values of P (high SNR) and a good choice of
equation coefficient vector am , it is
P hTm am |hTm am |
kam k
≤
|βm | = ≤
,
2
2
1 + P khm k
khm k
khm k
and accordingly for β̂m .
DISCO: Distributed Communication Systems
15 September 2013
8 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Noisy Channel Estimation - High SNR Approximation
We consider noisy estimates of the form
ĥm = hm + nm ,
where nm is the channel estimation error vector of
length L, assumed to be Gaussian distributed with
N (0, σ 2 IL×L ).
For high values of P (high SNR) and a good choice of
equation coefficient vector am , it is
P hTm am |hTm am |
kam k
≤
|βm | = ≤
,
2
2
1 + P khm k
khm k
khm k
and accordingly for β̂m .
DISCO: Distributed Communication Systems
15 September 2013
8 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Closed-Form Expressions
The error ε is then approximated by
ε̄ = kam k
1
−
kĥm k khm k
1
!
.
We define the following:
1
kam k
, k = L, λ = khm k,
khm k
σ
2
kam k2 + λσ x + β̄m
A(x) = −
,
2
2σ 2 x + β̄m
λkam k
.
B(x) =
σ x + β̄m
β̄m =
DISCO: Distributed Communication Systems
15 September 2013
9 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Closed-Form Expressions
The error ε is then approximated by
ε̄ = kam k
1
−
kĥm k khm k
1
!
.
We define the following:
1
kam k
, k = L, λ = khm k,
khm k
σ
2
kam k2 + λσ x + β̄m
A(x) = −
,
2
2σ 2 x + β̄m
λkam k
.
B(x) =
σ x + β̄m
β̄m =
DISCO: Distributed Communication Systems
15 September 2013
9 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Closed-Form Expressions
The probability density function (pdf) of ε̄2 is given by
"

√
k +1

√
2

k
ka
eA( x)
m

x))

k
k +2 I k −1 (B (
k
√
√
+1 2 −1

2
2
2
2σ
x ( x+β̄m )
λ



#


√

√
eA(− x)
2
x)) , 0 ≤ x ≤ β̄m
+ √
k +2 I k −1 (B (−
pε̄2 (x) =
2
2
x+
β̄
−
m
)
(






√

k
√

kam k 2 +1 eA( x)

2

x)) , β̄m
<x
k +2 I k −1 (B (
 k +1 k −1 √ √
2
2σ 2 λ 2
x( x+β̄m ) 2
The pdf of the computation rate loss ∆R̄ is also given in
closed form.
DISCO: Distributed Communication Systems
15 September 2013
10 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Closed-Form Expressions
The probability density function (pdf) of ε̄2 is given by
"

√
k +1

√
2

k
ka
eA( x)
m

x))

k
k +2 I k −1 (B (
k
√
√
+1 2 −1

2
2
2
2σ
x ( x+β̄m )
λ



#


√

√
eA(− x)
2
x)) , 0 ≤ x ≤ β̄m
+ √
k +2 I k −1 (B (−
pε̄2 (x) =
2
2
x+
β̄
−
m
)
(






√

k
√

kam k 2 +1 eA( x)

2

x)) , β̄m
<x
k +2 I k −1 (B (
 k +1 k −1 √ √
2
2σ 2 λ 2
x( x+β̄m ) 2
The pdf of the computation rate loss ∆R̄ is also given in
closed form.
DISCO: Distributed Communication Systems
15 September 2013
10 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Computation Rate Region of a 2-user Network
For given vectors hm = [0.9, 1.1] and am = [1, 1].
3,0
2
=0.01
2,5
Rate (bits per channel use)
2
=0.05
2
2,0
=0.15
1,5
2
=0.1
2
=0.15
1,0
Ideal Computational Rate
0,5
Computational Rate (simulation)
Computational Rate (approximation)
0,0
0
5
10
15
20
25
30
SNR (dB)
DISCO: Distributed Communication Systems
15 September 2013
11 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Computation Rate Region of a 4-user Network
For hm = [0.9, 1.1, 0.8, 1.2] and am = [1, 1, 1, 1].
2,0
2
=0.01
1,8
2
=0.05
Rate (bits per channel use)
1,6
1,4
1,2
1,0
2
=0.1
2
=0.15
0,8
0,6
0,4
Ideal Computational Rate
Computational Rate (simulation)
0,2
Computational Rate (approximation)
0,0
0
5
10
15
20
25
30
SNR (dB)
DISCO: Distributed Communication Systems
15 September 2013
12 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Rate Loss Percentage of a 2-user Network
For given vectors hm = [1, 1] and am = [1, 1].
60
SNR=20, 25, 30 dB
50
Rate Loss (%)
40
30
20
Simulation
10
Analytical Expression
0
0.00
0.05
0.10
0.15
0.20
2
DISCO: Distributed Communication Systems
15 September 2013
13 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Average Rate of a 2 user Network
For Rayleigh fading channels
3,5
Ideal Computation Rate with optimal
Computation Rate with optimal
Rate (bits per channel use)
3,0
a
m
Computation Rate with estimated
a
m
(sim.)
(sim.)
a
m
(sim.)
Approximate Computation Rate (optimal/estimated
a
m
)
2,5
2,0
1,5
1,0
0,5
0,0
0
5
10
15
20
25
30
SNR (dB)
DISCO: Distributed Communication Systems
15 September 2013
14 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The optimal coding vector
K. N. Pappi, G. K. Karagiannidis, and D. Toumpakaris, “Low Complexity
PHY-Layer Network Coding for Two-Way Compute-and-Forward Relaying”,
submitted to IEEE Wireless Communications and Networking Conference
(WCNC), Istanbul, Turkey, 6-9 April 2014.
The computation rate for the MMSE coefficient can be
written as

2 !−1 
T
P
h
a
1
m m
 kam k2 −
.
R(hm , am ) = log+
2
2
1 + P khm k2
DISCO: Distributed Communication Systems
15 September 2013
15 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The optimal coding vector
K. N. Pappi, G. K. Karagiannidis, and D. Toumpakaris, “Low Complexity
PHY-Layer Network Coding for Two-Way Compute-and-Forward Relaying”,
submitted to IEEE Wireless Communications and Networking Conference
(WCNC), Istanbul, Turkey, 6-9 April 2014.
The computation rate for the MMSE coefficient can be
written as

2 !−1 
T
P
h
a
1
m m
 kam k2 −
.
R(hm , am ) = log+
2
2
1 + P khm k2
DISCO: Distributed Communication Systems
15 September 2013
15 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The optimal coding vector
K. N. Pappi, G. K. Karagiannidis, and D. Toumpakaris, “Low Complexity
PHY-Layer Network Coding for Two-Way Compute-and-Forward Relaying”,
submitted to IEEE Wireless Communications and Networking Conference
(WCNC), Istanbul, Turkey, 6-9 April 2014.
The computation rate for the MMSE coefficient can be
written as

2 !−1 
T
P
h
a
1
m m
 kam k2 −
.
R(hm , am ) = log+
2
2
1 + P khm k2
DISCO: Distributed Communication Systems
15 September 2013
15 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The optimal coding vector
The optimal choice of am is given by
ao = arg min
2
a∈Z ,a6=0
aT G(h)a ,
P (hhT )
where G(h) = I − 1+P khk2 .
The above optimization has no analytical solution, but
it is algorithmically computed.
It can be mapped to the problem of finding the
shortest vector on a lattice with generator matrix
G(h).
The optimization is performed online for each channel
realization.
DISCO: Distributed Communication Systems
15 September 2013
16 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The optimal coding vector
The optimal choice of am is given by
ao = arg min
2
a∈Z ,a6=0
aT G(h)a ,
P (hhT )
where G(h) = I − 1+P khk2 .
The above optimization has no analytical solution, but
it is algorithmically computed.
It can be mapped to the problem of finding the
shortest vector on a lattice with generator matrix
G(h).
The optimization is performed online for each channel
realization.
DISCO: Distributed Communication Systems
15 September 2013
16 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The two-way relaying case: Reduction of candidate vectors
Theorem 2
The optimal equation coefficient vector for the TWRC is
either one of the vectors [1, 0]T , [0, 1]T , [1, 1]T , or its
elements are coprime numbers.
All pairs of coprime numbers (m, n) with m > n can be
arranged in a pair of disjoint complete ternary trees,
starting from (2, 1) or (3, 1) for even-odd or odd-odd pairs
respectively. The “children” of each vertex are generated as
(2m − n, m) , (2m + n, m) , (m + 2n, n) .
DISCO: Distributed Communication Systems
15 September 2013
17 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The two-way relaying case: Reduction of candidate vectors
Another criterion for the reduction of candidate vectors is
the inequality
kam k2 ≤ 1 + P khk2 .
For a given maximal transmitted power Pmax , and a margin
of probability b on the channel realizations we introduce
the criterion
kam k2 ≤ 1 + S 2 (Pmax , b) ,
where S (Pmax , b) is computed based on the channel
statistics.
DISCO: Distributed Communication Systems
15 September 2013
18 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The two-way relaying case: Reduction of candidate vectors
Another criterion for the reduction of candidate vectors is
the inequality
kam k2 ≤ 1 + P khk2 .
For a given maximal transmitted power Pmax , and a margin
of probability b on the channel realizations we introduce
the criterion
kam k2 ≤ 1 + S 2 (Pmax , b) ,
where S (Pmax , b) is computed based on the channel
statistics.
DISCO: Distributed Communication Systems
15 September 2013
18 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The two-dimensional space partition
ao = arg min
2
a∈Z ,a6=0
= arg min
2
a∈Z ,a6=0
kak2 + P kak2 khk2 − (hT a)2
kak2 + P kak2 khk2 sin2 (∠(a, h)) ,
√
√
g = [g1 , g2 ]T = [ P hmax , P hmin]T ,
DISCO: Distributed Communication Systems
15 September 2013
19 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
The two-dimensional space partition
Partition of (g1,g2) space using Algorithm 1
80
70
60
gmin (g2)
50
40
30
20
10
0
0
10
20
30
40
gmax (g1)
DISCO: Distributed Communication Systems
50
60
70
80
15 September 2013
20 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Optimal and Suboptimal Performance
Achievable Computation Rates at the Relay
3.5
2.5
2.0
1.5
1.0
R
c
(bits per channel use)
3.0
R
0.5
R
c
c
(optimum)
(low complexity)
0.0
0
5
10
15
20
25
30
SNR (dB)
DISCO: Distributed Communication Systems
15 September 2013
21 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Conclusions
Conclusions - Channel Estimation Errors
A formula for the achievable rate region for C&F
relays with Channel Estimation Errors was derived.
A tight approximation for Gaussian distributed
channel estimation errors was introduced and an
analytical expression for the pdf of the rate loss was
proposed.
Numerical and simulation results illustrate that C&F
is quite sensitive to channel estimation errors.
Future Work
Study of other models of channel estimation errors.
New criteria for the choice of βm and am by the C&F
relay.
DISCO: Distributed Communication Systems
15 September 2013
22 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Conclusions
Conclusions - TWRC PHY-Layer Network Coding
A low complexity PHY-Layer Network Coding
technique was developed for the TWRC, based on
look-up tables.
The look-up tables are constructed only once, offline,
based on the channel statistics and the maximum
transmission power of interest.
Although the network coding is suboptimal, it achieves
a performance remarkably close to the optimal one.
Future Work
Extension to complex systems.
Study of the method under channel estimation errors.
DISCO: Distributed Communication Systems
15 September 2013
23 / 24
Compute-and-Forward
Channel Estimation Errors
Physical Layer Network Coding
Conclusions
Thank you!
DISCO: Distributed Communication Systems
15 September 2013
24 / 24