BEP Analysis of OFDM Relay Links With Nonlinear Power Amplifiers
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8/10/2019 BEP Analysis of OFDM Relay Links With Nonlinear Power Amplifiers
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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
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8/10/2019 BEP Analysis of OFDM Relay Links With Nonlinear Power Amplifiers
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Introduction
Taneli Riihonen OFDM Relays with Nonlinear PAs 2 / 19
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Motivation
Taneli Riihonen OFDM Relays with Nonlinear PAs 3 / 19
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
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System Model
Taneli Riihonen OFDM Relays with Nonlinear PAs 4 / 19
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OFDM Signal with Nonlinear Distortion
Taneli Riihonen OFDM Relays with Nonlinear PAs 5 / 19
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]
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Two-Hop Amplify-and-Forward OFDM Relay Link
Taneli Riihonen OFDM Relays with Nonlinear PAs 6 / 19
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
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Multipath Channels
Taneli Riihonen OFDM Relays with Nonlinear PAs 7 / 19
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])
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BEP Analysis
Taneli Riihonen OFDM Relays with Nonlinear PAs 8 / 19
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BEP Derivation
Taneli Riihonen OFDM Relays with Nonlinear PAs 9 / 19
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]
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BEP Bounds
Taneli Riihonen OFDM Relays with Nonlinear PAs 10 / 19
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
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Discussion
Taneli Riihonen OFDM Relays with Nonlinear PAs 11 / 19
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Soft Limiter PA Model
Taneli Riihonen OFDM Relays with Nonlinear PAs 12 / 19
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
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Optimization of Back-off Factors (1)
Taneli Riihonen OFDM Relays with Nonlinear PAs 13 / 19
(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 ).
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Optimization of Back-off Factors (2)
Taneli Riihonen OFDM Relays with Nonlinear PAs 14 / 19
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
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Numerical Results (1)
Taneli Riihonen OFDM Relays with Nonlinear PAs 15 / 19
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
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Numerical Results (2)
Taneli Riihonen OFDM Relays with Nonlinear PAs 16 / 19
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
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Conclusion
Taneli Riihonen OFDM Relays with Nonlinear PAs 17 / 19
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Conclusion
Taneli Riihonen OFDM Relays with Nonlinear PAs 18 / 19
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
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Thank you!
Taneli Riihonen OFDM Relays with Nonlinear PAs 19 / 19