BEP Analysis of OFDM Relay Links With Nonlinear Power Amplifiers

8/10/2019 BEP Analysis of OFDM Relay Links With Nonlinear Power Amplifiers

1/19

BEP Analysis of OFDM Relay Links

with Nonlinear Power Amplifiers

Taneli Riihonen, Stefan Werner, Fernando Gregorio,

Risto Wichman

, and Jyri Hamalainen

Aalto University School of Science and Technology, FinlandUniversidad Nacional del Sur, Argentina

IEEE WCNC 2010, Sydney, Australia


2/19

Introduction

Taneli Riihonen OFDM Relays with Nonlinear PAs 2 / 19


3/19

Motivation


Goal:study the effect of nonlinear distortion in OFDM relay links

Background: cheap power amplifiers vs. high PAPR in OFDM

Focus:

Fixed infrastructure-based relay node Amplify and forward (AF) protocol

Frequency-domain processing

Base station

Relay nodeUser


4/19

System Model



5/19

OFDM Signal with Nonlinear Distortion


t rXt[n] Yr[n]

xt() xt()

htr()

PA

inve

rseDFT

DFT

CP

insertion

CP

removal

P/S

andD/A

A/D

andS/P

Signal model from transmittert {S, R}to receiverr {R, D} Tx: Xt[n]in frequency domain,xt()in time domain Nonlinear power amplifier (PA) static and memoryless:

xt() =ft(xt()) =Ktxt() + vt(), i.e., Xt[n] =KtXt[n] + Vt[n]

Power of distortion noise is2t = 1PtE{|Vt[n]|2} Multipath channel: htr()and in frequency domainHtr[n] Rx: Yr[n] =KtHtr[n]Xt[n] + HtrVt[n] + Wr[n]


6/19

Two-Hop Amplify-and-Forward OFDM Relay Link


S R DXS[n] XR[n]YR[n] YD[n]

xS() xS() xR() xR()

hSR() hRD()

PAPA

inve

rseDFT

inve

rseDFT

DFT

DFT

amp

lification

CPinsertion

CPinsertion

CP

removal

CP

removal

P/S

andD/A

P/S

andD/A

A/D

andS/P

A/D

andS/P

Parallel processing of subcarriers in frequency domain

Amplification by[n] =

PR(K2

S+2

S)PS|HSR[n]|

2+2R

End-to-end signal-to-noise ratio (SNR) becomes

[n] =

K2SSR[n]2SSR[n]+1

K2RRD[n]2RRD[n]+1

K2SSR[n]

2SSR[n]+1

+ K2

RRD[n]

2RRD[n]+1

+ 1where

SR[n] = PS|HSR[n]|2

2R

RD[n] = PR|HRD[n]|2

2D


7/19

Multipath Channels


The source and the relay are fixed nodes, and the destination is mobile

Base station

Relay nodeUser

Sourcerelay channelhSR(): stationary multipath components

Practical model: SR[n] = SR[n] =E {SR[n]} Not necessarily line-of-sight, i.e.,

SR[n]

=

SR[m]

Relaydestination channelhRD(): Rayleigh fading components

SNR distributionfRD[n](s) = (1/RD[n])exp(s/RD[n])


8/19

BEP Analysis



9/19

BEP Derivation


Reformulate the end-to-end SNR as [n] = 2 [n]RD[n][n]RD[n]+1in which

[n] =

1

2

K2SK2RSR[n]

(K2S+ 2S ) SR[n] + 1

[n] =

K2S

2R+

2S K

2R+

2S

2R

SR[n] + K

2R+

2R

(K2S+ 2S ) SR[n] + 1

Omitting few steps, average bit-error probability (BEP) iscalculated as

Pe[n] = 1

2

1

2[n]RD[n]

k=0

(1)k

k+ 32k!(k+

1

2 )

[n]

[n]

k+ 12

U

k+

3

2, 2,

1

[n]RD[n]


10/19

BEP Bounds


Linear PAs by substitutingKS =KR= 1and2S =

2R= 0

Asymptotic lower bounds

First hop limited by distortion: Pe[n] PSRe [n]withSR[n] = lim

SR[n][n] =

1

2

K2S K2R

K2S+ 2S

,

SR

[n] = limSR[n][n] =

K2S2R+

2S K

2R+

2S

2R

K2S+ 2S

Simpler bound by linear PAs: PSRe [n] 12 12

RD[n]RD[n]+2

Second hop limited by distortion:

Pe[n] PRDe [n] =1

2erfc

[n]

[n]

1

2erfc

SR[n]

2


11/19

Discussion



12/19

Soft Limiter PA Model


Transmittert

Clipping the amplitude of PA input signal:

|xt()|= ft(|xt()|) =

Att

|xt()|, |xt()| tPtAt

Pt, |xt()|> t

Pt

Adjustable input back-off2t PA factors in closed form:

Kt = At

t1 exp

2t +

t

2

erfc(t)2t =

A2t2t

1 exp 2t K2t


13/19

Optimization of Back-off Factors (1)


(S, R) = arg min(S,R)

Pe[n]subject toS>0,R>0

Can be decomposed as

S = arg minS

K2S SR[n]

2

S SR[n] + 1

, S >0

R = arg minR

Pe[n]S=S

, R >0

1.

1.51.5

25

1.525

1.55

1.55

1.55

1.5

5

1.575

1.575

1.575

1.575

1.6

1.6

1.6

1.6

1.6

1.625

1.625

1.625

1.625

1.62

5

1.65

1.65

1.65

1.65

1.

65

1.675

1.67

51.675

1.69

1.69

0 1 2 3 4 5 60

1

2

3

4

5

6

2S [dB]

2 R

[dB]

Fig. 2. Contour plot for log10`

Pe[n]

in terms of the input back-offfactors when SR[n] = 15dB, RD[n] = 20dB, and AS = AR = 1. Theminimal bit error probability P

e[n] = 2.0 102 is reached with 2

S =

2.85dB and 2R

= 2.18dB (marked with ).


14/19

Optimization of Back-off Factors (2)


6

6

5

5

4

4

3

3

2

2

1

1

0

0

1

1

2

2

3

3

4

4

5

5

6

6

0 5 10 15 20 25 300

5

10

15

20

25

30

SR[n] [dB]

RD

[n

]

[dB]

(a) 2S

[dB]

6 6 6

5 5 5

4 4 4

3 3 3

2 2 2

1 11

0 0

0

1

1

12

2

3

3

4

4

5

0 5 10 15 20 25 300

5

10

15

20

25

30

SR[n] [dB]

RD

[n

]

[dB]

(b) 2R

[dB]

0.5

0.7

5

0.75

0.75

1

1

1

1.25

1.25

1.25

1.5

1.5

1.75

1.75

2

2

2.2

5

2.25

2.5

2.75

0 5 10 15 20 25 300

5

10

15

20

25

30

SR[n] [dB]

RD

[n

]

[dB]

(c) log10`

Pe

[n]

Fig. 3. Contour plots for the optimized input back-off factors and the minimal bit error probability whenAS =AR = 1. The SNR pair (SR[n],RD[n])considered in Fig. 2 is marked with .

Numerical optimization validates the decomposition


15/19

Numerical Results (1)


0 5 10 15 20 25 30

Pe

[n]

[dB]

2S

= 6dB, 2R

= 0.5dB

2S

= 2dB, 2R

= 3dB

2S

= 2S

, 2R

= 2R

S linear, 2R

= 2R

2S

=2S

, R linear

Both PAs linear

101

102

103

104

Fig. 4. Average bit error probability whenSR[n] = , RD[n] = + 5dBand AS = AR = 1.

Input back-off adjustment is a trade-off between having closely

optimal BEP at low SNR or a low BEP floor at high SNR


16/19

Numerical Results (2)


0 5 10 15 20 25 30

Pe

[n]

SR[n] [dB]

2S

= 6dB,2R

= 0.5dB

2S

= 2dB, 2R

= 3dB

2S

= 2S , 2

R= 2

R

S linear, 2R

=2R

2S

= 2S

, R linear

Both PAs linear

101

102

Fig. 5. Average bit error probability in terms of the first hop SNR when thesecond hop SNR RD[n] = 20dB and AS = AR = 1.

0 5 10 15 20 25 30

Pe

[n]

RD[n] [dB]

2S

= 2S

, 2R

= 6dB

2S

= 2S

, 2R

= 2dB

2S

= 2S

, 2R

= 2R

S linear, 2R

=2R

2S

= 2S

, R linear

Both PAs linear101

102

Fig. 6. Average bit error probability in terms of the second hop SNR when thefirst hop SNR SR[n] = 15dB (hence

2S

= 2.85dB) and AS = AR = 1.

PA nonlinearity causes both SNR loss and higher BEP floors The performance is asymmetric: The second hop is more critical


17/19

Conclusion



18/19

Conclusion


Derivation of closed-form bit error probability expressions

Infrastructure-based amplify-and-forward OFDM relay link

The effect of nonlinear power amplifiers The performance with ideal linear PAs is a special case

The results are applicable to any memoryless power amplifier

Soft limiter model was selected for the numerical illustrations The adjustment of power amplifier input back-offs


19/19

Thank you!


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BEP Analysis of OFDM Relay Links With Nonlinear Power Amplifiers

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