### Mathematical model

In OFDM system, the analysis of the PAPR performance is similar to the OFDM system with single antenna. The PARP of the entire system is defined as the maximum of the PAPRs among all the transmit antennae [14, 15]. Equation 1 gives a mathematical representation of PARP where \(s\left( n \right)\) is peak-to-average power ratio of OFDM signals or the ratio between the maximum instantaneous power and its average power:

$$s\left( n \right) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt N }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sqrt N }$}}\mathop \sum \limits_{K = 0}^{N - 1} e^{{j2{\pi kn}/N}} .$$

(1)

*X*(*N*) represents the transmitted information in the *n*th subcarriers and *N* the number of subcarriers. Normally, the instantaneous output of the OFDM system has a large fluctuation compared to single carrier system.

Therefore, the peak-to-average power ratio is defined as

$${\text{PAPR}} = \hbox{max} |s\left( n \right)|^{2} /E|s\left( n \right)|^{2} ,$$

(2)

where \(\hbox{max} |s\left( n \right)|^{2}\) is the maximum power ratio of the OFDM and \(E|s\left( n \right)|^{2}\) is the average power ratio of the OFDM signal.

#### Windowing

Windowing consists of multiplying the signal by finite length window with amptitude that varies smoothly toward zero at the edges [15]. In this study, we considered three windows:

*Hamming window*

$$\begin{aligned} \text{Whamm} & = \begin{array}{ll} 0.54 + 0.46\cos \left( {\frac{2\uppi n}{N} - 1} \right) & \quad {{\text{for}} \ \, 0 \le n \ge N - 1} \end{array} \\ & = \begin{array}{ll} 0 & {\text{otherwise}} \end{array}, \end{aligned}$$

(3)

where *N* represents the width in samples of a discrete time of window function.

*Hanning window*

$$\text{Hann} = 0.5 + 0.5\cos \left( {\frac{{2{\uppi }n}}{N} - 1} \right)\quad \text{for} \;0 \le n \ge N - 1,$$

(4)

where *N* represents the width in samples of a discrete time of window function.

*Kaiser window*

$$\begin{aligned} {\text{Wkaiser}} &= \begin{array}{ll} I0 \left( \sqrt[\uppi \alpha]{1 - \left( {\frac{2n}{N - 1} - 1} \right)2} \right)\Big/I0(\uppi \alpha ); & \ \ 0 \le n \ge N - 1\end{array} \hfill \\ & = \begin{array}{ll} 0 & {\text{otherwise,}} \end{array} \end{aligned}$$

(5)

where *N* is the length of the sequence, *I*0 is the zeroth-order modified Bessel function and \(\alpha\) is the arbitrary non-negative real number that determines the shape of the window.

Considering the envelop of the OFDM signal to be Se(*t*) and the window function Wf, the function is expressed as [16, 17]

$${\text{Sp(}}t ) = \text{Se} (t ) * {\text{Wf}} .$$

(6)

The PAPR after peak windowing is expressed as

$$\text{PAPR} = \hbox{max} |\text{Sp}\left( t \right) |^{2} /E|\text{Sp}\left( t \right) |^{2} .$$

(7)

#### Clipping method

The clipping techniques used reduce PAPR by clipping the high peak of the OFDM signals by limiting the peak amplitude value to the threshold value [18]. Mathematically, clipping techniques can be defined as

$$B\left( x \right) = \left\{ {\begin{array}{*{20}c} {X \le C_{\text{L}} } \\ {C_{\text{L}} \text{ else}} \\ \end{array} } \right.,$$

(8)

where *B*(*x*), *C*_{L}, and *X* are clipped signal, clipping level, and input signal, respectively. The modified PAPR resulting from clipping can be expressed as

$$\text{PAPR} = C_{\text{L}}^{2} /E\left\{ {\left| {B\left( x \right)} \right|^{2} } \right\}.$$

(9)

Amplitude above the threshold value is clipped and information is lost. Due to the nonlinear operation, clipping causes the in-band distortion and out-band radiation. Due to the in-band distortion, BER performance is reduced [19, 20].