Practical Implementation of PN Scrambler for PAPR Reduction
Practical Implementation of PN Scrambler for PAPR Reduction
Practical Implementation of PN Scrambler for PAPR Reduction
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The probability that the power W > L occurs at least m=1<br />
time out <strong>of</strong> N samples is<br />
Pr{A occurs at least 1 time in N samples} =<br />
N ⎛ N ⎞ i N −i<br />
∑= ⎜ ⎟ ⋅ p q<br />
i 1 ⎝ i ⎠<br />
(19)<br />
In order to simplify the equation, note that the probability<br />
W > L occurs at least m=1 time out <strong>of</strong> N samples is the<br />
same as the complementary probability that exactly no<br />
samples (m=0) have W > L, which can be written as<br />
1-Pr{A occurs exactly 0 times out <strong>of</strong> N samples} =<br />
⎛ N ⎞ N<br />
N<br />
N<br />
1−<br />
g N ( 0)<br />
= 1−<br />
⎜ ⎟ ⋅ p q = 1−<br />
q = 1−<br />
( 1−<br />
p)<br />
= 1−<br />
F<br />
⎝ 0 ⎠<br />
0 N<br />
W ( L)<br />
(20)<br />
Thus, we now define the CCDF <strong>of</strong> the peak power per<br />
OFDM symbol as the probability that the peak power <strong>of</strong><br />
one sample out <strong>of</strong> N samples exceeds threshold L<br />
N<br />
⎛ ⎛ L ⎞⎞<br />
Fsym = Pr{<br />
PeakPowerOFDM<br />
_ Symbol > L}<br />
= 1−<br />
⎜1<br />
− exp⎜<br />
− ⎟⎟<br />
2<br />
⎝ ⎝ 2 ⋅σ<br />
⎠⎠<br />
(21)<br />
Using the <strong>PN</strong> scrambler to generate k OFDM symbols and<br />
then selecting the OFDM symbol with the lowest <strong>PAPR</strong><br />
<strong>for</strong> transmission reduces the peak power probability to p k .<br />
For example, if an OFDM symbol has the probability <strong>of</strong><br />
p=0.1 <strong>of</strong> exceeding threshold L, then the probability <strong>of</strong> k=2<br />
OFDM symbols exceeding threshold L is (0.1) 2 =0.01.<br />
This is true because <strong>of</strong> the statistical independence <strong>of</strong> the<br />
peak power between OFDM symbols generated by the <strong>PN</strong>-<br />
<strong>Scrambler</strong> (i.e. uncorrelated data sets). That is, if the<br />
probability <strong>of</strong> event S1 does not depend upon the outcome<br />
<strong>of</strong> S2 and visa versa, such events are said to be statistically<br />
independent and<br />
Pr(S , S )<br />
=<br />
1<br />
2<br />
2<br />
= Pr(S1)<br />
⋅ Pr(S2)<br />
= p ⋅ p p (22)<br />
For k statistically independent events, this becomes<br />
Pr(S , S ,..., S )<br />
=<br />
1<br />
2<br />
k<br />
k<br />
= Pr(S1)<br />
⋅Pr(S2<br />
) ⋅⋅⋅<br />
Pr(Sk<br />
) = p⋅<br />
p⋅<br />
⋅⋅<br />
p p (23)<br />
For k scrambling sequences, the new <strong>PAPR</strong> CCDF per<br />
OFDM symbol can now be <strong>for</strong>mally written as<br />
Pr<br />
{ <strong>PAPR</strong> k > L}<br />
OFDM _ Symbol sequences<br />
N<br />
⎛<br />
⎞<br />
⎜ ⎛ ⎛ L ⎞⎞<br />
= 1−<br />
⎜1−<br />
exp⎜<br />
− ⎟⎟<br />
⎟<br />
⎜<br />
2<br />
2 ⎟<br />
⎝ ⎝ ⎝ ⋅σ<br />
⎠⎠<br />
⎠<br />
k<br />
2 1<br />
σ =<br />
2<br />
(24)<br />
Substituting equation (13) into (21) yields<br />
F = 1 − F<br />
N<br />
(25)<br />
sym<br />
W<br />
4 <strong>of</strong> 7<br />
Rearranging, the sample-based power distribution is<br />
related to the OFDM symbol-based peak power<br />
distribution by<br />
( ) N<br />
1<br />
FW<br />
= 1− F<br />
(26)<br />
sym<br />
Next, we define the new <strong>PAPR</strong> distribution Fsym ˆ as the<br />
OFDM symbol-based peak-power distribution after k<br />
scrambling sequences<br />
k<br />
N k<br />
F ˆ<br />
sym = Fsym<br />
= ( 1 − FW<br />
)<br />
(27)<br />
Finally, in order to determine FW ˆ , which is the new <strong>PAPR</strong><br />
reduced sample-based power distribution after application<br />
<strong>of</strong> k scrambling sequences, we simply substitute Fsym ˆ into<br />
equation (26) relating the sample-based power distribution<br />
to the OFDM symbol-based peak power distribution,<br />
Fˆ<br />
W<br />
=<br />
1<br />
N<br />
N k<br />
( 1 − Fˆ<br />
) = ( 1 − ( 1 − F ) )<br />
sym<br />
W<br />
1<br />
N<br />
1<br />
N<br />
k N<br />
⎛<br />
⎞<br />
⎜ ⎛<br />
L ⎞<br />
1 ⎜ ⎛ ⎛ ⎞⎞<br />
⎟<br />
= ⎜ − 1 − ⎜1<br />
− exp⎜<br />
− ⎟ ⎟<br />
2 ⎟<br />
2<br />
⎟<br />
⎜ ⎜<br />
⎟ ⎟<br />
⎝ ⎝ ⎝ ⎝ ⋅σ<br />
⎠⎠<br />
⎠ ⎠<br />
(28)<br />
In the above equation, FW ˆ is the CDF <strong>of</strong> the <strong>PAPR</strong><br />
reduced sample-based distribution using k-scrambling<br />
sequences, which is the probability that each individual<br />
sample’s peak power is less than or equal to threshold<br />
level L using an IFFT size <strong>of</strong> N. It is more useful to<br />
evaluate the probability that each individual sample’s peak<br />
power is greater than threshold level L, which is the<br />
complementary CDF, CCDF=1-CDF.<br />
1<br />
k<br />
N<br />
N<br />
⎛<br />
⎞<br />
⎜ ⎛<br />
L ⎞<br />
CCDF(<br />
L,<br />
N,<br />
k)<br />
1 1 ⎜ ⎛ ⎛ ⎞⎞<br />
⎟<br />
= − ⎜ − 1−<br />
⎜1−<br />
exp⎜<br />
− ⎟ ⎟<br />
2 ⎟<br />
2<br />
⎟<br />
⎜ ⎜<br />
⎟ ⎟<br />
⎝ ⎝ ⎝ ⎝ ⋅σ<br />
⎠⎠<br />
⎠ ⎠<br />
(29)<br />
After normalization <strong>of</strong> the complex time-domain signal’s<br />
average power<br />
<strong>PAPR</strong><br />
CCDF<br />
1<br />
N k N<br />
= 1−<br />
⎜<br />
⎛1− ( 1−<br />
( 1−<br />
exp(<br />
− L)<br />
) ) ⎞ (30)<br />
( L,<br />
N,<br />
k)<br />
⎟<br />
⎝<br />
⎠<br />
It is important to note that due to the equivalence<br />
theorem, the analysis and linear signal processing <strong>of</strong><br />
baseband wave<strong>for</strong>m presented herein will be identical to<br />
the analysis <strong>of</strong> the bandpass wave<strong>for</strong>m with the exception<br />
<strong>of</strong> oversampling error. A rigorous discussion <strong>of</strong> the <strong>PAPR</strong><br />
measurement error due to Nyquist sampling versus<br />
continuous sampling can be found in [10]. Because most<br />
applications are typically concerned with the probability<br />
that a signal’s power exceeds a certain level L, instead <strong>of</strong><br />
using the CDF, the CCDF is typically plotted and is shown<br />
in Fig. 3 <strong>for</strong> FFT sizes N=64, N=128, N=256 and number<br />
<strong>of</strong> scrambling sequences k=1 (i.e. no <strong>PAPR</strong> reduction),<br />
k=8, and k=256. The plot shows both the theoretical<br />
curves obtained using equation (30) and the empirical<br />
curves obtained through computer simulations.