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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After evaluating the derivative with respect to k, the joint<br />
PDF is given as<br />
f<br />
W , k<br />
( L,<br />
N,<br />
k)<br />
N<br />
N<br />
⎛<br />
L ⎞ ⎛<br />
L ⎞<br />
⎜ ⎛ ⎛ ⎞⎞<br />
⎜ ⎛ ⎛ ⎞⎞<br />
− ln 1−<br />
⎜1−<br />
exp⎜−<br />
⎟⎟<br />
⎟ ⋅ 1−<br />
⎜1−<br />
exp⎜−<br />
⎟⎟<br />
⎟<br />
2<br />
⎜<br />
⎟ ⎜<br />
⎟<br />
⎝ ⎝ ⎝ 2 ⋅σ<br />
⎠⎠<br />
⎠ ⎝ ⎝ ⎝ 2 ⋅σ<br />
⎠⎠<br />
⎠<br />
= 2<br />
The probability in (31) can now be found by integrating<br />
the joint PDF over the range <strong>of</strong> k,<br />
( L,<br />
N,<br />
k)<br />
dk = p<br />
k<br />
(33)<br />
k m<br />
k > = − ∫ f<br />
=<br />
Pr( 256)<br />
1<br />
(34)<br />
W , k<br />
k =0<br />
Equation 34 can be solved <strong>for</strong> L to obtain<br />
⎛<br />
⎜ ⎛<br />
L = − ln⎜1<br />
− ⎜<br />
1−<br />
p<br />
⎜ ⎝<br />
⎝<br />
1<br />
1 N<br />
m<br />
⎞<br />
⎟<br />
⎠<br />
⎞<br />
⎟<br />
⎟<br />
⎟<br />
⎠<br />
(35)<br />
Equation (35) can be used to find the required <strong>PAPR</strong><br />
threshold level L <strong>for</strong> any FFT size N. The value m in (35)<br />
is the desired number <strong>of</strong> k-<strong>PN</strong> scrambling sequences not to<br />
exceed. The value p in (35) specifies the probability that a<br />
given OFDM symbol will not require more than m<br />
scrambling sequences in order to pass the objective <strong>PAPR</strong><br />
threshold level setting L. Evaluating (35) using p=10 -4 ,<br />
m=256, and N=64 yields the required <strong>PAPR</strong> threshold<br />
level setting L=2.98 or 4.74 dB.<br />
In order to find the average number <strong>of</strong> scrambling<br />
sequences, it is convenient to make use <strong>of</strong> the definition <strong>of</strong><br />
the expected value <strong>of</strong> a random variable,<br />
∞<br />
∫<br />
−∞<br />
X = E[<br />
X ] = x ⋅ f ( x)<br />
dx<br />
(36)<br />
The average number <strong>of</strong> scrambling sequences k becomes<br />
= ∞<br />
∫<br />
=<br />
⎟ ⎟⎟<br />
k ⎛<br />
N<br />
N<br />
k ⎞<br />
⎜ ⎛<br />
⎞ ⎛<br />
⎞<br />
⎜ ⎛ ⎛ L ⎞⎞<br />
⎟ ⎜ ⎛ ⎛ L ⎞⎞<br />
(37)<br />
k = k ⋅<br />
⎟<br />
⎜−<br />
ln 1−<br />
⎜1−<br />
exp⎜−<br />
⎟⎟<br />
⋅ 1−<br />
⎜1−<br />
exp⎜−<br />
⎟⎟<br />
dk<br />
2<br />
2<br />
⎜ ⎜<br />
⎟ ⎜<br />
⎟<br />
⎝ ⎝ ⎝ ⎝ 2⋅σ<br />
⎠⎠<br />
⎠ ⎝ ⎝ ⎝ 2⋅σ<br />
k 0<br />
⎠⎠<br />
⎠ ⎠<br />
After simplification,<br />
−1<br />
(38)<br />
k =<br />
N<br />
⎛<br />
⎞<br />
⎜ ⎛ ⎛ L ⎞⎞<br />
ln 1−<br />
⎜1−<br />
exp⎜<br />
− ⎟⎟<br />
⎟<br />
⎜<br />
2 ⎟<br />
⎝ ⎝ ⎝ 2⋅σ<br />
⎠⎠<br />
⎠<br />
For this example, the average number <strong>of</strong> <strong>PN</strong> scrambling<br />
sequences that will occur at this <strong>PAPR</strong> threshold level<br />
setting is<br />
( ( ( ) ) ) 27.8<br />
1<br />
64<br />
ln 1 1 exp 2.<br />
98<br />
=<br />
−<br />
k =<br />
(39)<br />
− − −<br />
Using 64-QAM modulation per subcarrier, the<br />
overhead v <strong>for</strong> this example is<br />
log 2 ( m)<br />
log ( 256)<br />
(40)<br />
2<br />
v =<br />
= = 0.<br />
028<br />
N ⋅ N ⋅ C 3<br />
bps 64 ⋅ 6 ⋅<br />
4<br />
where m is the number <strong>of</strong> sequences not to exceed, N is the<br />
number <strong>of</strong> subcarriers, Nbps is the number <strong>of</strong> bits per<br />
subcarrier based on the constellation size, and C is the<br />
FEC code rate.<br />
6 <strong>of</strong> 7<br />
In summary, in order to ensure that the OFDM<br />
transmitter does not exceed k=256 <strong>PN</strong> scrambling<br />
sequences with a probability <strong>of</strong> p=10 -4 , the peak power<br />
threshold level must be set to L=4.74 dB which will<br />
require approximately k=28 <strong>PN</strong> scrambling sequences on<br />
average. In other words, the system will have a <strong>PAPR</strong> less<br />
than or equal to 4.74 dB with a 99.9900% probability.<br />
Without the symbol scrambling (i.e. k=1), the <strong>PAPR</strong> is<br />
11.3 dB at p=10 -4 . Using the system parameters derived in<br />
this example, this technique results in an improvement <strong>of</strong><br />
6.5 dB with an insignificant overhead <strong>of</strong> 2.8%.<br />
4. PRACTICAL IMPLEMENTATION<br />
The <strong>PN</strong>-<strong>Scrambler</strong> was implemented as a <strong>PAPR</strong><br />
reduction technique <strong>for</strong> an IEEE 802.11a OFDM modem.<br />
Without application <strong>of</strong> the <strong>PAPR</strong> reduction, the CCDF<br />
curve <strong>for</strong> the IEEE 802.11a OFDM modem closely follows<br />
the k = 1 curve <strong>of</strong> Fig. 3, indicating that the IEEE 802.11a<br />
wave<strong>for</strong>m has a large <strong>PAPR</strong>.<br />
<strong>PAPR</strong> <strong>Reduction</strong> ON<br />
<strong>PAPR</strong> <strong>Reduction</strong> ON<br />
Back<strong>of</strong>f = 8 dB<br />
Back<strong>of</strong>f = 8 dB<br />
EVM = -40.34 dB<br />
EVM = -40.34 dB<br />
<strong>PAPR</strong> <strong>Reduction</strong> ON<br />
<strong>PAPR</strong> <strong>Reduction</strong> ON<br />
Back<strong>of</strong>f = 5 dB<br />
Back<strong>of</strong>f = 5 dB<br />
EVM = -33.44 dB<br />
EVM = -33.44 dB<br />
<strong>PAPR</strong> <strong>Reduction</strong> OFF<br />
<strong>PAPR</strong> <strong>Reduction</strong> OFF<br />
Back<strong>of</strong>f = 8 dB<br />
Back<strong>of</strong>f = 8 dB<br />
EVM = -34.35 dB<br />
EVM = -34.35 dB<br />
<strong>PAPR</strong> <strong>Reduction</strong> OFF<br />
<strong>PAPR</strong> <strong>Reduction</strong> OFF<br />
Back<strong>of</strong>f = 5 dB<br />
Back<strong>of</strong>f = 5 dB<br />
EVM = -25.40 dB<br />
EVM = -25.40 dB<br />
Figure 5. VSA EVM plots with and without<br />
<strong>PAPR</strong> reduction<br />
The OFDM signal from the output <strong>of</strong> the I/Q<br />
modulator was sent through a 10-Watt Stealth Microwave<br />
Class-A Power Amplifier (SM0825-40) into an Agilent<br />
Vector Signal Analyzer 89641A (VSA). The VSA<br />
demodulated the OFDM signal and the resulting<br />
constellations are shown in Fig. 5. The results show<br />
significant reduction in the Error Vector Magnitude<br />
(EVM) with <strong>PAPR</strong> reduction “on” versus with <strong>PAPR</strong><br />
reduction “<strong>of</strong>f.” Furthermore, the results show that at 8 dB<br />
back<strong>of</strong>f from the power amplifier’s one dB compression<br />
point (P1dB) with <strong>PAPR</strong> reduction turned-<strong>of</strong>f produces<br />
approximately the same EVM as 5 dB back<strong>of</strong>f when the<br />
<strong>PAPR</strong> reduction is turned-on. This 3 dB reduction in<br />
back<strong>of</strong>f provides twice the RF output transmit power, or<br />
equivalently, allows a 10-Watt amplifier to be used instead<br />
<strong>of</strong> a 20-Watt amplifier.