vahid meghdadi july 2009 - unilim.fr€¦ · vahid meghdadi xlim/c2s2 university of limoges power...

Cooperative techniques

Vahid MeghdadiJuly 2009

Vahid Meghdadi XLIM/C2S2 University of Limoges

Plan

� Amplify and Forward (AF) Scheme� Two-hop serial and parallel� Multi-hop serial and parallel� SNR calculations with MRC at the destination� SER calculation for high SNR� All Participate AF (AP-AF) versus Selection AF

(S-AF)

� Cooperative multiple access transmission using precoding vector


Simple A&F Scheme

P1

P2

N2

N1A

-The gain at the relay is A-The noises are CN(0,Ni)-Suppose the power of the constellation is normalized to unity

1- Hasna, M.O.; Alouini, M.-S., "End-to-end performance of transmission systems with relays over Rayleigh-fading channels," Wireless Communications, IEEE Transactions on , vol.2, no.6, pp. 1126-1131, Nov. 20032- A.Ribeiro, X. Cai, G. Giannakis, “Symbol Error probabilities for General Cooperative Links”, IEEE trans on commun., May 2005.

h1

h2

y1

x1

y2


System Model

( )1 1 1 1 1

2 2 1 1 1 1 2

y Ph x n

y Ah Ph x n n

= +

= + +

The SNR at each stage is defined as follows:2 2

1 1 2 21 2

1 2

P h P h

N Nγ γ≜ ≜

( )222 1 1 1P A P h N= +The transmitted power at the relay is

The SNR at the output will be:

2 222 1 1 1 222

1 22 1 21

A h p h

A h N N

γ γγγ γ

= =+ ++

And this is independent of A.


SNR at the Output

1 2 1 2

1 2 1 21γ γ γ γγγ γ γ γ

= ≈+ + +

1 2

1 1 1γ γ γ

≈ +

Therefore, a serial link can be imagined as the cascade of two resistances and the SNRs can be considered as the conductance of each link. The total conductance (SNR) can be found using the above formula.

1γ 2γ

γ

G1 G2

G

Approximation (high SNR):

Another way of approximation: ( )2 22 22 1 1 1 1 1P A P h N A P h= + ≈

If we replace this quantity in SNR calculation we will obtain the same equation as above:

1 2

1 1 1γ γ γ

≈ +


Optimum Gain A

The question is “what is the optimum gain to be applied in the relay ?”Supposing the approximation stated before, the SNR should be maximized, or the following function should be minimized by optimally distribution of total power:

1 2

1 1 1γ γ γ

≈ +

The constraint is : 1 2P P P cte+ = =

We obtain:

Considering that the power is distributed between the relay and the source:

1 2, (1 )P P P Pβ β= = −

1 2 1 22 2 2 2

1 1 2 2 1 2

1

(1 )

N N N N

P h P h P h P hγ β β= + = +

−

Derivation with respect to beta equal to zero:


Power Allocation

1 22 22 2

1 2

1 1 10

(1 )N N

P h P hβ γ β β∂ − = + = ∂ −

Simplifying the above equation gives:

2

2

2

2 2

1 2

1 2

h

N

h h

N N

β =

+

In the limit cases, if h2 goes to zero, beta is small, which means that the power should be used in the second link. In other words, it is not useful to expend power in the first link knowing that the second link cannot transmit the information.

Defining the normalized SNR as:2 2

1 21 2

1 2

and h h

N Nγ γ′ ′≜ ≜

2 2 1 1 21 2

2 11 2 1 2 1 2

and P

P P P PP

γ γ γ γβγγ γ γ γ γ γ

′ ′ ′ ′= ⇒ = = ⇒ =

′′ ′ ′ ′ ′ ′+ + +

2 1

1 2 2

1 NA

h h N≈ P

P

(this is obtained Supposing )

222 1 1P A P h≈


Power allocation

•If at the relay the information of the forwarding channel is not available (h2 is unknown), then h2 can be replaced by its mean and it is assumed that N1=N2.

2

1 2

1. { }

Ah mean h

≈2 2 2 (0,1)r ih h jh CN= + ∼

In the papers, a convenient solution is to choose the gain in order that the relay power to be equal to the original transmitted power. This is not the best solution !

2 12

1 1 1

PA

P h N=

+

|h2| is a Rayleigh distributed RV with 2 2

2

/22( ) h

h

hp h e σ

σ−=

The mean of |h2| is2 1

2 (3 / 2)2

σ πΓ =

2 0.5σ =

2

1

2A

hπ≈


Performance comparison

0 2 4 6 8 10 12 14 16 18 2010

-3

10-2

10-1

100

Es/N0

SE

R

optimum

mean h2papers


10-1

100

100

Performance comparison

Log 1/N0

Log SN

R

2

1

2A

hπ≈

2 1

1 2 2

1 NA

h h N≈

2 12

1 1 1

PA

P h N=

+


Adding Direct Path

P1

P2

N2

N1A

h1

h2

f

MRC can be used at the receiver. It means that we have two copies of the same signal x, and we use the weighting factors proportional to SNR of each branch. The result is a random variable z with the following SNR:

2

11 2

1 2 0

where z f f

P f

N

γ γγ γ γγ γ

= + =+

Here the approximation for SNR of relayed signal is used (the one is ignored).

The distribution of SNRf is exponential.

G1 G2

Gz

Gr


Power allocation

�Since the original signal is received twice, the optimal power allocation is not the same as before. The optimal solution can be found by derivation of the total SNR with respect to beta.

Compare with matlab simulation the three previous cases for this case.


Symbol Error Rate Calculations

The SER is calculated by averaging over the pdf of SNR.

0

( ) ( )eP AQ k p dγγ γ γ∞

= ∫

Note that in AWGN, for many constellations, like MPSK and QAM, the symbol error rate can be calculated asWhere k depends on constellation.For example for BPSK, A=1,k=2 ; QPSK: A=2,k=1;

MPSK:

( )s err sP AQ kγ− =

Note that the Q function goes to zero very fast with increasing gamma. So the integral has only significant values at small SNRs. It means the Pe is well dimensioned by low SNRs.

2 2 sins err sP QM

πγ− ≈


Symbol Error Rate Approximation

ˆ

0 0

ˆ ˆ ˆ( ) ( ) ( ) ( )Q k p d Q k p dγ γγ γ γ γγ γ γ∞ ∞

=∫ ∫

It is clear that the PDF of γ around zero is the most important because for larger γ the product of the two functions can be ignored.

It is proved1 that for a system with diversity order t+1, the first term in the Taylor series of will be

1- Zhengdao Wang; Giannakis, G.B., "A simple and general parameterization quantifying performance in fading channels," Communications, IEEE Transactions on , vol.51, no.8, pp. 1389-1398, Aug. 2003

ˆ ˆ( )pγ γ

ˆ ˆ ˆ ˆ( ) ( )tp a oγ γ γ γ= +

Where has the “same distribution” as with unity average.ˆ /γ γ γ= γ


Symbol Error Rate Approximation

This approximation can be plugged in the integral and we obtain:

ˆ

2

ˆ

0 0

ˆ ˆ ˆ ˆ ˆ( ) ( )ˆ2

k

teQ k p d a d

k

γγ

γγγ γ γ γ γπ γγ

−∞ ∞

≈∫ ∫

This was obtained using the approximation for Q function:

2 /21( )

2xQ x e

x π−≤

Changing the variable:

The error probability is finally:

ˆ2

kx

γγ=

11

(2 1)( )

t

s tl

AaP l

kγ +=

≤ −∏

And using

1

( 1/ 2) (2 1)2

t

tl

t lπ

=

Γ + = −∏


Taylor Series of ( )pγ γ

As seen before ˆ

ˆ 0

ˆ( )1ˆ!

t

t

pa

tγ

γ

γγ

=

∂=

∂

11

(2 1)1

(0)!

t

t

ls t t

lp

P Ak t

γ

γ=

+

− ∂=

∂

∏Therefore

Here the is replaced by , that’s why the at the denominator was simplified.

γ̂ γ γ


Cooperative Configuration

( )1 2

11 2

(2 1)1 1

(0) (0) (0) (0)! f

t

t

ls t t

lp

P A A p p pk t k

γγ γ γγ

=+

− ∂= = +

∂

∏

P1

P2

N2

N1A

h1

h2

f

So we should study now the distribution of1 2

1 2z f

γ γγ γγ γ

= ++

is a random variable and zγ ( )1 2

(0) (0) (0) (0)z

f

z

pp p pγ

γ γ γγ∂

= +∂


Multihop Serial Case

R0 RN+1

0 1

0 1 1 1 00

...

... ...

NN

i ii

γ γ γγγ γ γ γ γ− +

=

=∑ 0 1

1 1 1 1...

Nγ γ γ γ= + + +or

The error probability can be approximated by [1]:0

1(0)

2 i

N

ei

P Pk γ

=

≈ ∑

G0 G1

totG γ=

GN

1- A.Ribeiro, X. Cai, G. Giannakis, “Symbol Error probabilities for General Cooperative Links”, IEEE trans on commun., May 2005.


Parallel Two-Hop Relays

R1

R2

RM

1

i i

i i

Mg h

z fi g h

γ γγ γ

γ γ=

= ++∑

**

2 21

MRC i

i

Mi i i

D RiD R

h A gfz y y

σ σ=

⇒ = +

∑

1

( 0)(0) (0) (0)z

f g hi i

M Mz

Miz

pp p pγ

γ γ γ

γγ =

∂ = = + ∂ ∏

tot totG γ=

igih

1/i iA g=


Selection AF (S-AF) scheme [1][2]

� In this scheme only the “best relay” is used for relaying

� The best relay is the one who

2 2

, ,2 2

, ,

,1 ,2

,1 ,2

arg max

1

arg max1

i s i i

s i i d

ii s i i

s i i d

i i

ii i

g E h E

N Ni

g E h E

N N

γ γγ γ

=+ +

=+ +

R1

R2

RM

igih

1/i iA g=

1- Yi Zhao; Adve, R.; Teng Joon Lim, "Symbol error rate of selection amplify-and-forward relay systems," Communications Letters, IEEE , vol.10, no.11, pp.757-759, November 2006

2- Zhao, Y.; Adve, R.; Lim, T.J., "Improving amplify-and-forward relay networks: optimal power allocation versus selection," Wireless Communications, IEEE Transactions on , vol.6, no.8, pp.3114-3123, August 2007

f


AP-AF versus S-AF

� In S-AF the required parallel (orthogonal) channels are less� Just 2 channels are required versus

N+1 in AP-AF

� In S-AF the exact knowledge of the channels is not required.

� A full diversity of m+1 is obtained in both cases.

� Because the best channel uses all the power, better performance can be obtained

12

!1

mSeAP

e

Pm

P m

+ = + m is the number of relays


SER calculation of S-AF

2

0 max1

S i ir

ii i

α β γγ α γα γ β γ

= ++ +

Where 2 22

0 0, , , and 1/s i i s i i if E g E h E Nα α β γ= = = =

It can be shown that 1

0 1( )

( )1R

m mi iS i r

rFm

λ λ ξ γγγ

+=

Γ

+ = +

∏

where λ0 is the exponential distribution parameter of α0 :0 0 0( ) exp( )f x xα λ λ= −

And the same for and i iλ ξ0 1

1

(2 1)! ( )

( 1)!(2 )

m

i iis m

mP

m k

λ λ ξγ=

+

+ +≈

+∏

So full diversity of m+1 is achieved.


Conclusion over AF transmission

� The outage probability is improved considerably

� Independent channels obtained

� Use of cooperative network is justified in mobile communications

� AF allows full diversity and simple relays

� S-AF give better performance with simpler receiver

� Tight upper bound for high SNR can be analytically obtained


Plan

� Amplify and Forward (AF) Scheme� Two-hop serial and parallel� Multi-hop serial and parallel� SNR calculations with MRC at the destination� SER calculation for high SNR� All Participate AF (AP-AP) versus Selection AF

(S-AF)

� Cooperative multiple access transmission using precoding vector


Multiple access cooperative transmission[1]

� One transmitter and several receivers.� Each receiver is interested on its own data.

BS

R1

R2

MS

MS

MS

R antennas

N Mono-AntennaMobile stations

High SNR / High ratelink

11h

2h N

H. Meghdadi, V. Meghdadi, J.P. Cances, “Cooperative Multiple Access Transmission Using Precoding vectors” EUSIPCO 2009.

1x

2x

1Y

2Y

YN

Beam forming


Problem

� Cancelling multi-antenna and multi-relay interference

� Maximizing the SNR at destinations� This is done using linear filtering: weighting vector at

relay nodes� The BS send a vector of symbols of size N.� Each relay, by one channel use, send a sequence of

size R over its R antennas.� Assumption

� Each relay knows all the channels (CSI)� Each relay only know its own channel


System model

Transmitted information symbols

Signals at the output of the relays (of size R where R is the number of TX antennas)

1,...,j N=Signal at the j th mobile.

are the precoding vectors to be optimized.W ij


Interference cancellation

All the contributions of all other symbols except for j th should be cancelled out.

First case: each relay knows all the CSI

Second case: each relay knows only its own channel

Therefore:

Must be a real positive number


Precoding vector calculations

Definitions:

Using these notations, the second case interference cancelation becomes

Kroneckerproduct

Row wiseKronecker

product


Transforming on real equation

with real entries.Therefore we have

The equation can be written as


Coherent addition

Respecting this constraint, the received signal at each MS will be:

The coefficient of sj should be a positive real, so :

Using matrix representation:


Power constraint and objective function

This is the optimization constraint which limits the transmitted power of each relay.

Therefore the function to be maximized is the SNR at mobile stations with the following constraints

Instead, we can minimize the power but fix the SNR. This can be better optimized using Lagrange multipliers.

So the function to be minimized is:

Note: We are looking for the solutions independent of s.


SNR constraint

� One possibility is to maximize the sum of the all the SNRs at all MS.

� Another possibility is to make MS’s SNR equal and to maximize it

� Another possibility can be to let the SNRs to be different. This will let us to have the MS with more privilege.


Solution

� Case: flexible constraints over SNR at MSs � Solution : Lagrange multipliers

� Special case : SNRs at all MS are forced to be equal� Solution : Lagrange multipliers method gives the same answer as

pseudo inverse method

So we require . Normalizing by using g=1:


Solution …

� The solution is

Note, H must be full rank : / 2R N≥

� In this solution, all the relays had to know all the CSI. If each relay knows only its own channel to MSs :

� De-normalizing

Note, Hi must be full rank : R N≥


Performance

� The received signal is(a vector of size N)

� SNR will be

� It is difficult to obtain the distribution of γ, so we calculate an upper bound for BEP for high SNR.


BEP calculations

� In general, the symbol error probability can be calculated as:

where

� Replacing and using Taylor series

for and supposing a diversity order of t+1


Simulation

Gd=3

Gd=4

Gd=5

Independent relays: Gd = LR - L(N-1)Complete CSI : Gd = LR - (N-1)


Simulation

Complete CSI and L=2, N=2, and R=3

Diversity 5


Simulation

L=2, N=2, and R=3

Gd=4

Gd=5


Simulation

L=1


Conclusion

� Optimum precoding vectors calculated for different scenarios, thanks to Lagrange multipliers.

� The special case where the constraint that all the SNR at the mobile stations are the same reduces to Moore-Penrose pseudo-inverse.

� The case where all relays knows all the CSI is considered which gives a single large matrix pseudo-inversion.

� The case where each mobile knows only its own channel is also considered which is less performing but easier to implement practically.

� Semi analytic tight upper bound is evaluated for general case.

vahid meghdadi july 2009 - unilim.fr€¦ · vahid meghdadi xlim/c2s2 university of limoges power...

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