Practical Implementation of PN Scrambler for PAPR Reduction

PRACTICAL IMPLEMENTATION OF PN SCRAMBLER FOR PAPR REDUCTION IN OFDM 

SYSTEMS FOR RANGE EXTENSION AND LOWER POWER CONSUMPTION 

Christopher Moffatt 

Harris Corporation, Melbourne, FL, USA 

Christopher.Moffatt@Harris.com 

ABSTRACT 

Orthogonal Frequency Division Modulation (OFDM) is an 

efficient multi-carrier modulation scheme. It is used in 

many wireless communication systems due to its 

robustness towards fading channel behavior and a relative 

ease of implementation coming from computationally 

efficient Inverse Fast Fourier Transforms (IFFT). 

However, the OFDM waveform has a very large Peak-to- 

Average Power Ratio (PAPR). Therefore, to avoid clipping 

plus nonlinear distortion, the Power Amplifier (PA) needs 

to be backed off significantly. This paper proposes a novel 

implementation and analysis of a Pseudorandom Noise 

(PN) scrambling technique allowing a significant 

reduction of the OFDM signal’s PAPR. The theoretical 

basis for the technique is presented, and a particular 

implementation of the technique for the IEEE 802.11a 

standard is discussed. Both simulations and measurements 

demonstrate significant benefits of the proposed technique 

in longer range wireless communications applications that 

use OFDM. 

1. INTRODUCTION 

Multi-carrier waveforms for digital communications 

require a summation of multiple frequency-spaced singlecarriers 

prior to transmission through a Power Amplifier 

(PA). Orthogonal Frequency Division Multiplexing 

(OFDM) is a multi-carrier technique that utilizes hardware 

efficient IFFT to modulate each individual subcarrier with 

a Quadrature Amplitude Modulated (QAM) symbol and 

sum the subcarriers together to produce a single timedomain 

waveform. The time-domain waveform produced 

by addition of the independently modulated subcarriers has 

a very large Peak-to-Average Power Ratio (PAPR). As a 

result, the average power into the PA must be backed-off 

to avoid clipping of the time-domain signal peaks. 

Clipping of the signal significantly increases the in-band 

noise (IBN) and the out-of-band noise (OBN) which 

adversely increases the bit-error rate (BER) and the 

adjacent channel interference (ACI), respectively. The 

large backoff necessary to avoid clipping and provide 

operation in the linear region of the PA requires higher 

power amplifiers at the transmitter and greatly increases 

system DC power consumption. For example, using 64 

sub-carriers, a 40 dBm power amplifier could require 

about 10 dB of backoff for a clipping probability of 

978-1-4244-2677-5/08/$25.00 ©2008 IEEE 

1 of 7 

Ivica Kostanic, Ph.D. 

Florida Institute of Technology, Melbourne, FL, USA 

Kostanic@FIT.edu 

p=0.01%. Therefore, instead of operating at 40 dBm (10 

Watts), the amplifier must be operated at 30 dBm (1 Watt) 

average power. Alternatively, in order to transmit at 40 

dBm, a 50 dBm (100 Watt) amplifier would be required. 

In practice however, large PAPR values occur infrequently, 

and the backoff is typically reduced down to a 

compression level that causes an acceptable amount of 

distortion. Still, a large portion of the DC power 

consumed is wasted solely to maintain the linearity of the 

output PA stage. Thus, the use of PAPR reduction to 

allow less PA backoff is imperative to provide lower 

power requirements and extended battery operation. 

The outline of the paper is as follows. Section 2 

presents a brief overview of PAPR reduction techniques. 

Section 3 discusses the proposed PN scrambling-based 

PAPR reduction method. Section 4 describes a practical 

implementation of the PN Scrambler applied to an IEEE 

802.11a OFDM physical layer in an FPGA-based Software 

Defined Radio (SDR). Section 5 presents the example 

power savings applied to practical amplifiers. Finally, 

some conclusions are drawn in Section 6. 

2. PAPR REDUCTION METHODS 

The PAPR of a waveform may be described [1] as 

2 

x( 

t) 

max 

PAPR = (1) 

Pavg 

where Pavg is the average power of the waveform. 

In practical OFDM systems, the PAPR may be 

reduced using one or a combination of several techniques. 

The techniques may be divided into three major categories. 

The first category employs various methods of nonlinear 

signal distortion such as hard clipping [2], soft clipping [3], 

companding [4], or pre-distortion [5]. Generally speaking, 

the nonlinear distortion techniques are simple to 

implement. However, many do not work well in cases 

where the OFDM sub-carriers are modulated with higherorder 

modulation schemes. In such scenarios, the 

Euclidian distance between the symbols is relatively small 

and the additional noise introduced by the PAPR reduction 

causes significant performance degradation. 

The second category for PAPR reduction employs 

various coding methods. The coding techniques [6] have 

an advantage of being distortionless and the PAPR 

reduction is most commonly achieved by eliminating 

symbols having large PAPR. However, to obtain an 

appreciable level of PAPR reduction, high redundancy

codes need to be used and as a result, the overall efficiency 

of transmission becomes reduced. 

Finally, the third category is based on OFDM symbol 

scrambling and selection of the sequence that produces 

minimum PAPR. The pre-scrambling techniques [7-8] 

achieve good PAPR reduction but they require multiple 

FFT transforms and somewhat higher processing power. 

The method presented in this paper belongs to the third 

category of the PAPR reduction techniques. It uses 

conveniently chosen Pseudorandom Noise (PN) sequences 

applied to the input data bit stream. The method is very 

easy to realize in the software or hardware environment 

which is very important if the PAPR needs to be 

implemented in Application Specific Integrated Circuits 

(ASIC). In such a scenario, the PN-Scrambler may be 

implemented by the addition of external FPGA and DSP 

hardware to the Commercial Off-The-Shelf (COTS) 

ASICs. As a result, one obtains cost efficient and reliable 

hardware solutions. 

3. OVERVIEW OF THE PN-SCRAMBLER 

TECHNIQUE FOR PAPR REDUCTION 

3.1. System overview 

Block diagrams of the transmitter and receiver 

implementing the proposed PN-Scrambler are presented in 

Figs. 1 and 2, respectively. 

A/D Real 

A/D Imag 

Figure 1. PN-Scrambler Tx Block Diagram 

DC 

Offset 

DC 

Offset 

Subcarrier 

Demapper 

I / Q 

Imbalance 

I / Q 

Imbalance 

Channel 

Estimate 

Acquisition / Synchronization 

AGC 

Receive (Demodulation) Path 

Timing 

Detect 

Carrier 

Phase 

and 

Timing 

Drift 

Correction 

Coarse 

Freq 

Soft 

Bit 

Decisions 

Timing 

Fine 

Freq 

Deinterleaver 

and 

Convolutional 

decoder 

Sample 

Buffer 

Guard 

Interval 

Removal 

Descrambler 

FFT 

Data Bits 

Output 

Figure 2. PN-Scrambler Rx Block Diagram 

As seen in Fig. 1, two additional elements are added to 

a typical OFDM transmitter. The first element is the 

PAPR scrambler, and the second one is the PAPR 

threshold compare block. The PN-Scrambler utilizes a 

Maximal-Length Linear Feedback Shift Register (ML- 

LFSR) with log2(k) = l taps in order to produce k = 2 l -1 

uncorrelated unique sets of data from the same input 

sequence. The k unique sets of data are used to generate k 

independent identically distributed (i.i.d.) OFDM symbols. 

A block of Nb bits comprising one OFDM symbol is 


scrambled and passed along for Forward Error Correction 

(FEC) coding, interleaving, modulation, symbol mapping 

and IFFT. 

In any given OFDM system, Nb is a function of the 

number of subcarriers, the modulation scheme applied to 

each subcarrier, and the coding rate. By examining each 

individual sample coming out of the IFFT, the PAPR 

threshold comparator determines if the scrambler has 

achieved a desired PAPR on a symbol-by-symbol basis. If 

the PAPR of the symbol is below a desired threshold, then 

the data is passed along towards the RF stage of the 

transmitter. However, if the PAPR is still high, the data is 

scrambled with a different phase of the ML-LFSR’s PN 

sequence. Since this technique operates on the input bit 

stream, it is essentially independent of the OFDM 

modulation and may be adapted to any particular scenario. 

The receiver presented in Fig. 2 is a typical OFDM 

receiver that needs to perform the tasks of down 

conversion, channel estimation, and decoding. The only 

additional task required by the PN-Scrambler PAPR 

reduction technique is descrambling of the data at the 

receiver output. To perform descrambling, the receiver 

has to “know” the phase of the ML-LFSR used on the 

transmission side. This phase is embedded in the data 

stream. For example, the first l bits of the OFDM symbol 

may carry the information on the ML-LFSR phase. 

3.2. Analytical Performance Characterization 

A practical implementation of the PN-Scrambler 

PAPR reduction technique requires selection of several 

parameters. These parameters are defined as follows. 

1. Number of scrambling sequences (k) - defined as the 

number of PN sequences produced by the ML- 

LFSR. Each sequence is Nb bits long. 

2. PAPR threshold (L) – defined as the maximum 

PAPR for the OFDM symbol. This value is used by 

the PAPR threshold comparator block in order to 

discard OFDM symbols with PAPR greater than L. 

3. IFFT size / number of sub-carriers (N) – defined as 

the number of the non-zero orthogonal subcarriers 

per OFDM symbol. 

4. Average latency ( k ) – defined as the average 

number of scrambling attempts per OFDM symbol 

in order to pass the threshold level L. 

5. Probability of clipping (p) – probability that the 

PAPR exceeds the threshold level L after k 

scrambling attempts. 

6. PN scrambler overhead ( v ) – defined as the ratio of 

the number of bits required to represent the phase of 

the ML-LFSR to the number of bits per OFDM 

symbol Nb. 

In any actual design, the above parameters allow different 

tradeoffs. The subsequent section highlights some of these 

design trades.

3.3. PAPR per OFDM Sample after Application of k- 

PN Scrambling Sequences 

The goal of the analysis presented in this section is to 

find the analytical PAPR probability distribution of the 

PN-scrambled OFDM signal on a sample-by-sample basis. 

This allows the PAPR to be compared with other sampled 

time-domain signals and techniques. Define the complex 

baseband signal as 

S = X + jY , where j = −1 

(2) 

From the central limit theorem, the OFDM signal will have 

a Gaussian probability density function (PDF) for both the 

real X and imaginary Y time-domain signals due to the 

addition of multiple complex exponentials with random 

amplitude A and phase θ. 

N −1 

N −1 

⎛ 2π 

nk ⎞ 

⎛ 2πnk 

⎞ ⎛ 2πnk 

⎞ (3) 

∑ Ak ⋅ exp⎜ 

j ⋅ + jθ 

k ⎟ = ∑ Ak 

⋅ cos⎜ 

+ θ k ⎟ + j ⋅ Ak 

⋅ sin⎜ 

+ θ k ⎟ 

n= 

0 ⎝ N ⎠ n= 

0 ⎝ N ⎠ ⎝ N ⎠ 

Then, the magnitude R, is 

R + 

The magnitude of the complex baseband signal can be 

approximated as Rayleigh distributed, since the real and 

imaginary components X and Y are independent Gaussian 

random variables with zero mean and equal variances σ 2 . 

Interestingly, the original derivation of this density 

function by Lord Rayleigh in 1880 was applied to the 

similar application of finding the envelope of the sum of 

many sine waves of different frequencies. The Rayleigh 

PDF is given by [9] as 

2 

⎧ r ⎛ r ⎞ 

⎪ exp 

≥ 

⎨ ⎜ 

⎜− 

⎟ r 0 

f ( r) 

= 2 

2 

(5) 

R σ ⎝ 2 ⋅σ 

⎠ 

⎪ 

⎩ 0 r < 0 

The PDF of the power signal of a complex Gaussian signal 

is derived using a transformation of random variables, 

where it is desired to find the power random variable 

2 

W = R 

(6) 

Using a transformation of random variables 

dr 

fW ( w) 

= f R ( r) 

⋅ 

(7) 

dw 

The complex 1 power PDF can be written as 

⎧⎛ 

1 ⎞ ⎛ w ⎞ 

⎪⎜ 

⎟ ⋅ exp⎜ 

− ⎟ w ≥ 0 

f ( w) 

= 

2 

2 

⎨ 

(8) 

W ⎝ 2 ⋅σ 

⎠ ⎝ 2 ⋅σ 

⎠ 

⎪ 

⎩ 0 

w < 0 

The complex power CDF of a complex signal with 

Gaussian real and imaginary components is given by 

w 

⎛ 1 ⎞ ⎛ u ⎞ 

Pr( W ≤ w) 

= FW 

( w) 

= ∫ ⎜ ⎟ ⋅exp⎜ 

− ⎟du (9) 

2 

2 

0 ⎝ 2⋅σ 

⎠ ⎝ 2⋅σ 

⎠ 

Since this is an indeterminate integral, it cannot be 

represented in closed-form solution. However, it is still 

2 2 

= X Y 

(4) 

1 The term “complex” power refers to the OFDM complex baseband 

signal in order to avoid confusion from the power PDF of a real variable. 


possible to find the complex power CDF 2 by taking 

advantage of the Rayleigh CDF. 

2 

W = R 

2 2 

= X + Y 

(10) 

The complex power CDF is given by, 

FW ( w) 

= Pr( R ≤ + w) 

− Pr( R ≤ − w) 

(11) 

Then, the probability that the power is less than or equal to 

some value w is simply the probability that the magnitude 

is between the values of + w and − w . The CDF for the 

magnitude R is the Rayleigh CDF 

2 

⎧ ⎛ r ⎞ 

⎪1 

− exp 

⎨ ⎜ 

⎜− 

⎟ 

Pr( R ≤ r) 

= F ( ) = 

2 

R r 

⎝ 2 ⋅σ 

⎠ 

⎪ 

⎩ 0 

r ≥ 0 (12) 

r < 0 

Substituting equation (12) into equation (11) and using the 

fact that FR ( − w) 

= 0 since FR ( r) 

= 0 when r < 0 

⎧ ⎛ w ⎞ 

⎪1 

− exp⎜ 

− 

Pr( W ≤ w) 

= F ( ) = 

2 ⎟ 

W w ⎨ ⎝ 2 ⋅σ 

⎠ 

⎪ 

⎩ 0 

w ≥ 0 

w < 0 

(13) 

The peak power per OFDM symbol is measured as the 

largest power sample out of the N corresponding complex 

samples. It is desired to find the probability that the peak 

power of at least one out of N samples exceeds a 

predetermined peak power threshold level L. We start by 

defining event A as the event that the complex timedomain 

sample’s power w exceeds threshold L. We want 

to find the probability that event A occurs at least m=1 

time out of the N OFDM symbol samples. Since each 

time-domain sample is i.i.d., the Bernoulli trials can be 

useful here 

Pr{A occurs m times} = 

⎛ N ⎞ m N −m 

g N ( m) 

= ⎜ ⎟ ⋅ p q (14) 

⎝ m ⎠ 

⎛ N ⎞ 

N! 

⎜ ⎟= 

N Cm 

= 

(15) 

⎝ m ⎠ m! 

( N − m)! 

Using the Binomial distribution, we can write 

Pr{A occurs at least m times in N samples}= 

N 

N ⎛ N ⎞ i N −i 

∑ g N ( i) 

= ∑⎜ 

⎟ ⋅ p q 

(16) 

i= 

m 

i= 

m ⎝ i ⎠ 

where p is the probability that a single complex timedomain 

sample exceeds L, defined as 

2 { x + jy > L} 

= 1− 

F ( L) 

p = Pr 

(17) 

w 

q W 

= 1− p = F ( L) 

(18) 

2 The probability distribution function, or cumulative distribution 

function (CDF), is defined by [10] to be the probability of the event that 

the observed random variable W is less than or equal to the allowed 

value w. In this case, W is the random variable and the value w will 

later represent a power threshold level and will be substituted with the 

variable L.

The probability that the power W > L occurs at least m=1 

time out of N samples is 

Pr{A occurs at least 1 time in N samples} = 

N ⎛ N ⎞ i N −i 

∑= ⎜ ⎟ ⋅ p q 

i 1 ⎝ i ⎠ 

(19) 

In order to simplify the equation, note that the probability 

W > L occurs at least m=1 time out of N samples is the 

same as the complementary probability that exactly no 

samples (m=0) have W > L, which can be written as 

1-Pr{A occurs exactly 0 times out of N samples} = 

⎛ N ⎞ N 

N 

N 

1− 

g N ( 0) 

= 1− 

⎜ ⎟ ⋅ p q = 1− 

q = 1− 

( 1− 

p) 

= 1− 

F 

⎝ 0 ⎠ 

0 N 

W ( L) 

(20) 

Thus, we now define the CCDF of the peak power per 

OFDM symbol as the probability that the peak power of 

one sample out of N samples exceeds threshold L 

N 

⎛ ⎛ L ⎞⎞ 

Fsym = Pr{ 

PeakPowerOFDM 

_ Symbol > L} 

= 1− 

⎜1 

− exp⎜ 

− ⎟⎟ 

2 

⎝ ⎝ 2 ⋅σ 

⎠⎠ 

(21) 

Using the PN scrambler to generate k OFDM symbols and 

then selecting the OFDM symbol with the lowest PAPR 

for transmission reduces the peak power probability to p k . 

For example, if an OFDM symbol has the probability of 

p=0.1 of exceeding threshold L, then the probability of k=2 

OFDM symbols exceeding threshold L is (0.1) 2 =0.01. 

This is true because of the statistical independence of the 

peak power between OFDM symbols generated by the PN- 

Scrambler (i.e. uncorrelated data sets). That is, if the 

probability of event S1 does not depend upon the outcome 

of S2 and visa versa, such events are said to be statistically 

independent and 

Pr(S , S ) 

= 

1 

2 

2 

= Pr(S1) 

⋅ Pr(S2) 

= p ⋅ p p (22) 

For k statistically independent events, this becomes 

Pr(S , S ,..., S ) 

= 

1 

2 

k 

k 

= Pr(S1) 

⋅Pr(S2 

) ⋅⋅⋅ 

Pr(Sk 

) = p⋅ 

p⋅ 

⋅⋅ 

p p (23) 

For k scrambling sequences, the new PAPR CCDF per 

OFDM symbol can now be formally written as 

Pr 

{ PAPR k > L} 

OFDM _ Symbol sequences 

N 

⎛ 

⎞ 

⎜ ⎛ ⎛ L ⎞⎞ 

= 1− 

⎜1− 

exp⎜ 

− ⎟⎟ 

⎟ 

⎜ 

2 

2 ⎟ 

⎝ ⎝ ⎝ ⋅σ 

⎠⎠ 

⎠ 

k 

2 1 

σ = 

2 

(24) 

Substituting equation (13) into (21) yields 

F = 1 − F 

N 

(25) 

sym 

W 


Rearranging, the sample-based power distribution is 

related to the OFDM symbol-based peak power 

distribution by 

( ) N 

1 

FW 

= 1− F 

(26) 

sym 

Next, we define the new PAPR distribution Fsym ˆ as the 

OFDM symbol-based peak-power distribution after k 

scrambling sequences 

k 

N k 

F ˆ 

sym = Fsym 

= ( 1 − FW 

) 

(27) 

Finally, in order to determine FW ˆ , which is the new PAPR 

reduced sample-based power distribution after application 

of k scrambling sequences, we simply substitute Fsym ˆ into 

equation (26) relating the sample-based power distribution 

to the OFDM symbol-based peak power distribution, 

Fˆ 

W 

= 

1 

N 

N k 

( 1 − Fˆ 

) = ( 1 − ( 1 − F ) ) 

sym 

W 

1 

N 

1 

N 

k N 

⎛ 

⎞ 

⎜ ⎛ 

L ⎞ 

1 ⎜ ⎛ ⎛ ⎞⎞ 

⎟ 

= ⎜ − 1 − ⎜1 

− exp⎜ 

− ⎟ ⎟ 

2 ⎟ 

2 

⎟ 

⎜ ⎜ 

⎟ ⎟ 

⎝ ⎝ ⎝ ⎝ ⋅σ 

⎠⎠ 

⎠ ⎠ 

(28) 

In the above equation, FW ˆ is the CDF of the PAPR 

reduced sample-based distribution using k-scrambling 

sequences, which is the probability that each individual 

sample’s peak power is less than or equal to threshold 

level L using an IFFT size of N. It is more useful to 

evaluate the probability that each individual sample’s peak 

power is greater than threshold level L, which is the 

complementary CDF, CCDF=1-CDF. 

1 

k 

N 

N 

⎛ 

⎞ 

⎜ ⎛ 

L ⎞ 

CCDF( 

L, 

N, 

k) 

1 1 ⎜ ⎛ ⎛ ⎞⎞ 

⎟ 

= − ⎜ − 1− 

⎜1− 

exp⎜ 

− ⎟ ⎟ 

2 ⎟ 

2 

⎟ 

⎜ ⎜ 

⎟ ⎟ 

⎝ ⎝ ⎝ ⎝ ⋅σ 

⎠⎠ 

⎠ ⎠ 

(29) 

After normalization of the complex time-domain signal’s 

average power 

PAPR 

CCDF 

1 

N k N 

= 1− 

⎜ 

⎛1− ( 1− 

( 1− 

exp( 

− L) 

) ) ⎞ (30) 

( L, 

N, 

k) 

⎟ 

⎝ 

⎠ 

It is important to note that due to the equivalence 

theorem, the analysis and linear signal processing of 

baseband waveform presented herein will be identical to 

the analysis of the bandpass waveform with the exception 

of oversampling error. A rigorous discussion of the PAPR 

measurement error due to Nyquist sampling versus 

continuous sampling can be found in [10]. Because most 

applications are typically concerned with the probability 

that a signal’s power exceeds a certain level L, instead of 

using the CDF, the CCDF is typically plotted and is shown 

in Fig. 3 for FFT sizes N=64, N=128, N=256 and number 

of scrambling sequences k=1 (i.e. no PAPR reduction), 

k=8, and k=256. The plot shows both the theoretical 

curves obtained using equation (30) and the empirical 

curves obtained through computer simulations.

Probability of PAPR > x dB 

10 -1 

10 -2 

10 -3 

10 -4 

10 -5 

CCDF Curve 

Analytical PAPR CCDF 

Empirical Prob k: 1 FFT SIZE: 64 

Derived Prob k: 1 FFT SIZE: 64 

















10 

0 2 4 6 8 10 12 

-6 

x dB 

Figure 3. PAPR probability distribution per sample using 

k-PN scrambling sequences 

Figure 3 shows that the PAPR probability given in 

equation (30) matches the simulated data very closely. 

The figure shows a significant reduction in PAPR due to 

application of the k-PN scrambling sequences. For 

example, for N=64 subcarriers, the figure indicates the 

probability that the PAPR exceeds L=4.5 dB reduces from 

p=6.0% at k=1 down to p=0.01% at k=256. In order to 

ensure that the PAPR is less than L at p=0.01% of the time 

for conventional OFDM (i.e. k=1), the equivalent PAPR 

threshold is L=9.6dB. If the OFDM signal is input into an 

ideal amplifier that clips the signal when it is larger than 

the saturation point and it is desired to not clip the signal 

more than p=0.01% of the time, then application of the 

k=256 scrambling sequences requires 9.6-4.5=5.1 dB less 

backoff. Often a larger amount of in-band noise and outof-band 

noise can be tolerated and therefore a higher 

clipping probability is acceptable. Therefore, in a typical 

system, less than 5.1 dB performance is gained due to 

application of the k=256 scrambling sequences. 

Probability of PAPR > L dB 

10 0 

10 -1 

10 -2 

10 -3 

10 -4 

10 -5 

CCDF Curve Vs. Threshold L and k-PN Scrambling Sequences 

k=256 

Increasing k Scrambling Sequences 

10 

0 2 4 6 8 10 12 

-6 

L dB 

Figure 4. CCDF Curve Probability PAPR>L, multiple k- 

PN scrambling sequences overlaid (k=1:1:256) 

k=1 


The CCDF curves in Fig. 4 show rapid PAPR reduction 

during the first few k scrambling sequences. After about 

k=8, the PAPR reduction begins to approach diminishing 

returns. For example, at k=128 and k=129 the probability 

p=Pr{PAPR>L}=0.01% occurs at threshold L=4.7786 dB 

and L=4.7753 dB, respectively. This results in a small 

improvement of ΔL=0.0033 dB, which is insignificant 

compared to ΔL=1.4 dB from k=1 to k=2. 

3.4. Practical Example 

As a practical example, consider the following design 

problem. An OFDM transmitter with N=64 subcarriers 

must be designed. As part of the design problem, the 

transmitter must achieve a low latency and therefore use 

no more than k=256 scrambling sequences per OFDM 

symbol. Each OFDM symbol is compared to a threshold 

level L. If the OFDM symbol’s peak power exceeds the 

threshold level L, it is re-generated using the PN- 

Scrambler until the symbol’s peak power is smaller than L. 

As soon as the OFDM symbol has a peak power less than 

L, it is stored in a packet buffer where it waits to be 

transmitted. Once the desired number of “symbols per 

packet” is stored in the packet buffer, the packet is 

immediately transmitted out the digital-to-analog converter 

(DAC). Since the PN-Scrambler in this example is only 

l=8-bits, the ML-LFSR can only generate k=2 8 -1=255 

possible unique OFDM symbols that represent the same 

data. Since the transmitter already has a data whitener, an 

additional bypass mode is added to provide a total of 

k=255+1=256 possible unique OFDM symbols that 

represent the same data. If any of the symbols exceed the 

k=256 scrambling sequences, the symbol with the lowest 

PAPR out of the 256 will be selected for transmission. 

However, this means that one or more of the symbols in 

the packet buffer did not pass the threshold level L and 

may be clipped by the DAC or PA. Therefore, in order to 

ensure that this does not happen often, we desire a very 

low probability of 1 in 10,000 OFDM symbols requiring 

more than k=256 PN-Sequences, or equivalently, p=10 -4 . 

The problem is to determine the value of the threshold 

level L and the average number of k-PN scrambling 

sequences that will be generated as a result of the 

corresponding threshold level setting. 

We start by noting the latency requirement and let 

m=256, the number of sequences not to exceed. For this 

example, 

−4 

p = Pr( k > m) 

= Pr( k > 256) 

= 10 (31) 

Both L and N are predetermined constants in the 

transmitter and receiver and k is the random variable, 

f 

W , k 

( L, 

N, 

k) 

dF 

= 

W , k 

( L, 

N, 

k) 

dk 

k 

⎛ 

N ⎞ 

⎜ ⎛ 

L ⎞ ⎟ 

d ⎜ ⎛ ⎛ ⎞⎞ 

⎟ 

⎜1− 

1− 

⎜1− 

exp⎜− 

⎟ 2 ⎟ ⎟ 

⎜ ⎜ 

⎟ ⎟ 

⎝ ⎝ ⎝ ⎝ 2⋅σ 

⎠⎠ 

⎠ 

= 

⎠ 

dk 

(32)

After evaluating the derivative with respect to k, the joint 

PDF is given as 

f 

W , k 

( L, 

N, 

k) 

N 

N 

⎛ 

L ⎞ ⎛ 

L ⎞ 

⎜ ⎛ ⎛ ⎞⎞ 

⎜ ⎛ ⎛ ⎞⎞ 

− ln 1− 

⎜1− 

exp⎜− 

⎟⎟ 

⎟ ⋅ 1− 

⎜1− 

exp⎜− 

⎟⎟ 

⎟ 

2 

⎜ 

⎟ ⎜ 

⎟ 

⎝ ⎝ ⎝ 2 ⋅σ 

⎠⎠ 

⎠ ⎝ ⎝ ⎝ 2 ⋅σ 

⎠⎠ 

⎠ 

= 2 

The probability in (31) can now be found by integrating 

the joint PDF over the range of k, 

( L, 

N, 

k) 

dk = p 

k 

(33) 

k m 

k > = − ∫ f 

= 

Pr( 256) 

1 

(34) 

W , k 

k =0 

Equation 34 can be solved for L to obtain 

⎛ 

⎜ ⎛ 

L = − ln⎜1 

− ⎜ 

1− 

p 

⎜ ⎝ 

⎝ 

1 

1 N 

m 

⎞ 

⎟ 

⎠ 

⎞ 

⎟ 

⎟ 

⎟ 

⎠ 

(35) 

Equation (35) can be used to find the required PAPR 

threshold level L for any FFT size N. The value m in (35) 

is the desired number of k-PN scrambling sequences not to 

exceed. The value p in (35) specifies the probability that a 

given OFDM symbol will not require more than m 

scrambling sequences in order to pass the objective PAPR 

threshold level setting L. Evaluating (35) using p=10 -4 , 

m=256, and N=64 yields the required PAPR threshold 

level setting L=2.98 or 4.74 dB. 

In order to find the average number of scrambling 

sequences, it is convenient to make use of the definition of 

the expected value of a random variable, 

∞ 

∫ 

−∞ 

X = E[ 

X ] = x ⋅ f ( x) 

dx 

(36) 

The average number of scrambling sequences k becomes 

= ∞ 

∫ 

= 

⎟ ⎟⎟ 

k ⎛ 

N 

N 

k ⎞ 

⎜ ⎛ 

⎞ ⎛ 

⎞ 

⎜ ⎛ ⎛ L ⎞⎞ 

⎟ ⎜ ⎛ ⎛ L ⎞⎞ 

(37) 

k = k ⋅ 

⎟ 

⎜− 

ln 1− 

⎜1− 

exp⎜− 

⎟⎟ 

⋅ 1− 

⎜1− 

exp⎜− 

⎟⎟ 

dk 

2 

2 

⎜ ⎜ 

⎟ ⎜ 

⎟ 

⎝ ⎝ ⎝ ⎝ 2⋅σ 

⎠⎠ 

⎠ ⎝ ⎝ ⎝ 2⋅σ 

k 0 

⎠⎠ 

⎠ ⎠ 

After simplification, 

−1 

(38) 

k = 

N 

⎛ 

⎞ 

⎜ ⎛ ⎛ L ⎞⎞ 

ln 1− 

⎜1− 

exp⎜ 

− ⎟⎟ 

⎟ 

⎜ 

2 ⎟ 

⎝ ⎝ ⎝ 2⋅σ 

⎠⎠ 

⎠ 

For this example, the average number of PN scrambling 

sequences that will occur at this PAPR threshold level 

setting is 

( ( ( ) ) ) 27.8 

1 

64 

ln 1 1 exp 2. 

98 

= 

− 

k = 

(39) 

− − − 

Using 64-QAM modulation per subcarrier, the 

overhead v for this example is 

log 2 ( m) 

log ( 256) 

(40) 

2 

v = 

= = 0. 

028 

N ⋅ N ⋅ C 3 

bps 64 ⋅ 6 ⋅ 

4 

where m is the number of sequences not to exceed, N is the 

number of subcarriers, Nbps is the number of bits per 

subcarrier based on the constellation size, and C is the 

FEC code rate. 


In summary, in order to ensure that the OFDM 

transmitter does not exceed k=256 PN scrambling 

sequences with a probability of p=10 -4 , the peak power 

threshold level must be set to L=4.74 dB which will 

require approximately k=28 PN scrambling sequences on 

average. In other words, the system will have a PAPR less 

than or equal to 4.74 dB with a 99.9900% probability. 

Without the symbol scrambling (i.e. k=1), the PAPR is 

11.3 dB at p=10 -4 . Using the system parameters derived in 

this example, this technique results in an improvement of 

6.5 dB with an insignificant overhead of 2.8%. 

4. PRACTICAL IMPLEMENTATION 

The PN-Scrambler was implemented as a PAPR 

reduction technique for an IEEE 802.11a OFDM modem. 

Without application of the PAPR reduction, the CCDF 

curve for the IEEE 802.11a OFDM modem closely follows 

the k = 1 curve of Fig. 3, indicating that the IEEE 802.11a 

waveform has a large PAPR. 

PAPR Reduction ON 


Backoff = 8 dB 


EVM = -40.34 dB 

EVM = -40.34 dB 





EVM = -33.44 dB 

EVM = -33.44 dB 

PAPR Reduction OFF 




EVM = -34.35 dB 

EVM = -34.35 dB 





EVM = -25.40 dB 

EVM = -25.40 dB 

Figure 5. VSA EVM plots with and without 

PAPR reduction 

The OFDM signal from the output of the I/Q 

modulator was sent through a 10-Watt Stealth Microwave 

Class-A Power Amplifier (SM0825-40) into an Agilent 

Vector Signal Analyzer 89641A (VSA). The VSA 

demodulated the OFDM signal and the resulting 

constellations are shown in Fig. 5. The results show 

significant reduction in the Error Vector Magnitude 

(EVM) with PAPR reduction “on” versus with PAPR 

reduction “off.” Furthermore, the results show that at 8 dB 

backoff from the power amplifier’s one dB compression 

point (P1dB) with PAPR reduction turned-off produces 

approximately the same EVM as 5 dB backoff when the 

PAPR reduction is turned-on. This 3 dB reduction in 

backoff provides twice the RF output transmit power, or 

equivalently, allows a 10-Watt amplifier to be used instead 

of a 20-Watt amplifier.

5. POWER SAVINGS 

The power consumption of communication systems is 

typically determined by the power requirements of the 

final power amplifier. From the CCDF curves given in Fig. 

3, it can be observed that the PN-Scrambler technique 

provides considerable reduction of the PAPR. As a result, 

the average power of the PA may be set closer to its 

maximum power level. For a Class-A Linear PA, a 3 dB 

lower P1dB-point typically corresponds to a reduction of 

DC power consumption by half (see Table 1 for examples). 

Table 1- Class-A Power Amplifiers 

(Courtesy of Stealth Microwave) 

Class-A Linear Frequency RF Output Power RF Output DC Voltage Current Power 

Power Amplifier Range (MHz) P1dB (dBm) Power (Watts) (V) (Amps) (Watts) 

SM1720-37H 1700-2000 37 5.0 12 1.9 22.8 

SM1720-41L 1700-2000 41 12.6 12 5.5 66 

SM1720-44L 1700-2000 44 25.1 12 10 120 

SM1720-48L 1700-2000 48 63.1 12 16 192 

SM1720-50L 1700-2000 50 100.0 12 19 228 

As a further example of the benefits, consider a design 

that utilizes a PA with an RF output power of 25.1 Watts 

(P1dB = 44 dBm). According to Table 1, this corresponds 

to DC power consumption of 120 W. With 3 dB of PAPR 

reduction, a 12.6 W (41 dBm) PA which consumes 66 W 

of DC power can be utilized to transmit the 

communications waveform with the same average output 

power. This directly results in a 54 W power savings. 

Additionally, the DC-DC power converter used to supply 

the 12 V to the PA is typically about 80% efficient. 

Therefore, to power the 25.1 W PA requires the system to 

supply about 150 W of DC power. The same system with 

the PN-Scrambler PAPR reduction technique applied 

requires 82.5 W to supply the 12.6 W PA. In summary, 

the total system DC power saved from the PAPR reduction 

is 67.5 W, which results in 45% power savings. 

6. CONCLUSION 

The PN-Scrambler technique was presented in this 

paper and its performance was accurately characterized. 

All of the analytical equations derived in this paper were 

verified against simulated data. The PN-Scrambler was 

shown to provide exceptional PAPR reduction for a low 

number of subcarriers. For example, given a PAPR 

requirement that the peak power must not exceed L more 

than once in 1000 samples (i.e. p=10 -3 or 0.1%) and by 

using k=256 PN scrambling sequences, it was shown that 

the achievable PAPR level is L=4.3 dB for N=64 

subcarriers, L=5.2 dB for N=128, and L=5.9 dB for N=256. 

Compared to conventional OFDM with N=64 subcarriers 

and PAPR level L=8.4 dB at p=10 -3 , this technique 

provides an improvement of ΔL=4.1 dB. 

The PN-Scrambler technique does not require 

significant modification of the transmitter chain and 

virtually no modification of the receiver chain. On the 

transmitter side, all that is needed is some external 


hardware and on the receiver side, the same ML-LFSR. 

This makes the technique very attractive for many 

practical applications where low-cost, robust solutions are 

desired. 

7. ACKNOWLEDGEMENTS 

The authors would like to express a sincere 

appreciation to Harris Corporation and Wireless Center of 

Excellence at Florida Institute of Technology for support 

of the research presented in this paper. 

8. REFERENCES 

[1] R. van Nee and R. Prasad, OFDM for Wireless Multimedia 

Communications. 1st ed., Norwood, MA: Artech House 

Publishers, 2000. 

[2] D. Wulich, and L. Goldfeld, “Reduction of Peak Factor in 

Orthogonal Multicarrier Modulation by Amplitude Limiting 

and Coding,” IEEE Transactions on Communications, Vol. 

47, No. 1, January 1999. 

[3] A. D. S. Jayalath and C. Tellambura, “Peak-to-Average 

Power ratio of IEEE 802.11a PHY layer Signals,” 6th 

International Symposium on Digital Signal Processing for 

Communication System DSPCS'02, (Sydney-Manly), pp.31- 

36, January 2002. 

[4] W. Xianbin, T. Tjhung, and C. S. Ng, “Reduction of Peak-to- 

Average Power Ratio of OFDM System Using a 

Companding Technique,” IEEE Transactions and 

Broadcasting, Vol. 45, No. 3, September 1999. 

[5] A. N. D’Andrea., V. Lottici, and R. Reggiannini, “Nonlinear 

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Fading Channels,” IEEE Transactions on Communications, 

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[6] K. G. Paterson, and V. Tarokh, “On the Existence and 

Construction of Good Codes with Low Peak-to-Average 

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[7] S. H. Muller, R. W. Bauml, R. F. H. Fischer, and J. B. Huber, 

“OFDM with Reduced PAPR by Multiple Signal 

Representation,” Annals of Telecommunications, Vol. 52, 

No. 1-2, pp 58-67, Fen 1997. 

[8] K. D. Choe, S. C. Kim, and S. K. Park, “Pre-scrambling 

methods for PAPR Reduction in OFDM Communication 

Systems,” IEEE Transactions on Consumer Electronics, 

Vol.50, No. 4, November 2004. 

[9] G. R. Cooper and C. D. McGillem, Probabilistic Methods of 

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the Peak-to-Average Power of OFDM,” IEEE Transactions 

on Communications, Vol. 49, No. 2, February 2001.

Practical Implementation of PN Scrambler for PAPR Reduction

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