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Nonlinear PAPR reducers, such as clipping and companding techniques, are some simple methods used to reduce the Peak-to-Average Power Ratio (PAPR). In this paper, assuming that the baseband OFDM signal is characterized as a band-limited complex Gaussian process, we investigate the PAPR distribution of an OFDM signal when it is passed through a nonlinear PAPR reducer. The obtained PAPR distribution depends on the nonlinear function which characterizes the PAPR reducer. Later in this paper, we apply the obtained PAPR distribution in the clipping case. The comparisons made between the proposed distribution and that obtained thanks to computer simulations show good agreement.

Orthogonal Frequency Division Multiplexing (OFDM) is an attractive modulation technique for the next generation of high bit rate wireless transmission due to its high robustness to multipath fading and its great simplification of channel equalization [

But, only few papers are dealing with PAPR distribution analysis at the output of PAPR reducers. Some researchers proposed a PAPR distribution analysis when there is unequal power allocation between carriers [

The remainder of this paper is organized as follows: Section 2 briefly introduces nonlinear PAPR reducers. In Section 3, the PAPR distribution is analyzed through its Complementary Cumulative Distribution Function (CCDF). Then in Section 4 results of previous analysis is applied to Soft Enveloppe Clipping reducer. In this section, we provided some results which show good agreement between simulation and theoretical expressions. Finally in Section 5, a conclusion is drawn.

Let, be the baseband equivalent time-domain OFDM signal. can be written as

where is the OFDM magnitude, is the OFDM phase and is the OFDM symbol period.

The PAPR of may be defined as

where is the signal average power.

In nonlinear PAPR reducers (clipping, companding techniques), the data signal PAPR is reduced by a nonlinear function as shown in

Now, let us suppose the nonlinear function that characterizes the nonlinear PAPR reducer shown in

where is nonlinear positive function also called function for PAPR reduction.

In the literature, it is customary to use the Complementary Cumulative Distribution Function (CCDF) of the PAPR as a performance criterion. Let us consider and the discrete-time signals at the Nyquist rate of the OFDM signal and its PAPR reduced version respectively. For a large number of subcarriers, the OFDM envelope converges to a Rayleigh envelope distribution. Therefore, the probability density function (PDF) of the OFDM envelope can be expressed as

where is the mean power OFDM signal.

Using (4), it was shown in [

where, is the number of samples per OFDM symbol period. This PAPR CCDF expression has been proved for the first time by R. van Nee and A. de Wild in [

In the same way as (5), we show that the PAPR distribution of the ouput signal could be approximated by Equation (6):

Using (2), it can be shown that, and Equation (6) becomes

Equation (7) shows that, the expression of depends on the function for PAPR reduction. In the following section of this paper, an exact expression of is given in the soft envelop clipping’s case [

In this section, in order to illustrate the theoretical results obtained for nonlinear PAPR reducers, we consider one nonlinear PAPR reducer which is commonly studied in the literature: The Soft Envelop Clipping (SEC) [

The nonlinear function of SEC is expressed as

where is the magnitude threshold and commonly known as clipping threshold.

It is shown in [

where is the Dirac impulse. From (9), we show that,

where is the clipping ratio (CR) and is the output-to-input average power ratio defined as

where is the function for PAPR reduction defined in (8) and is the PDF of the OFDM signal expressed in (4).

Substituting (10) into (7), we show that, the expression of for SEC is expressed as

when becomes great and tends to infinity, then tends to and expression 12 is equal to classical expression 5 of the CCDF at the input of the clipping.

In this paper, assuming that the baseband OFDM signal is characterized as a band-limited complex Gaussian process, we have investigated the PAPR distribution of an OFDM signal at the output of a nonlinear PAPR reducer. The obtained PAPR distribution has been applied in the clipping case, which is a well-known example of nonlinear PAPR reducer used for OFDM PAPR reduction.

The comparisons made between the proposed PAPR

distribution at the output of clipping and with that obtained thanks to computer simulations show good agreement.