Performance Analysis of Communication System with Nonlinear Power Amplifier

Abstract

A communication system with nonlinear Power Amplifier (PA) is considered. The PA nonlinearity forces the designer of communication system to choose proper Back-Off, which influences three major parameters of the communication system: (i) power efficiency, (ii) Adjacent Channel Power Ratio (ACPR), and (iii) bit rate expressed by Mutual Information (MI). Having these three parameters, we propose to evaluate the performance of a communication system with nonlinear PA using the fact that ACPR is dictated by the regulator/communication standard. In what follows we propose to find power efficiency and MI as a function of ACPR. To illustrate the usefulness of the proposed method, the Solid State Power Amplifier (SSPA), and two linearized versions of SSPA are considered. The first version is a perfectly linearized PA using pre-distortion resulting with Soft Envelope Limiter (SEL), and the second is the Feed-Forward (FF) architecture based on SSPA. For each of the PAs ACPR, power efficiency and MI are found. While the analysis of SSPA and SEL is mainly based on the existing literature, the analysis of FF may be considered as a new contribution. From the results of the analysis it is concluded that linearization improves the overall performance of communication system. It is also shown that the pre-distortion method (SEL) shows better performance than FF.

Appendix: PDF of the Received Signal

Since s and n are statistically independent, the PDF of $y, {p_y}\left( {u,v} \right)$, is just a convolution of the PDFs of s and n [32]:

$$ \begin{aligned} {p_y}\left( {u,v} \right) =& {p_s}\left( {u,v} \right)*{p_n}\left( {u,v} \right) \\ =& \int\limits_{{\tau _u}} {\int\limits_{{\tau _v}} {{p_s}\left( {{\tau _u},{\tau _v}} \right)} } {p_n}\left( {{\tau _u} - u,{\tau _v} - v} \right)d{\tau _u}d{\tau _v}. \end{aligned} $$

(41)

For convenience, we convert the Cartesian coordinates of (9) to polar coordinates using $u = r\cos \left( \theta \right)$ and $v = r\sin \left( \theta \right)$; therefore we can rewrite (9) as

$$ H\left( y \right) = - \int\limits_{r \ge 0} {\int\limits_\theta {{p_y}\left( {r,\theta } \right)} } {\log _2}\left( {{p_y}\left( {r,\theta } \right)} \right)rdrd\theta $$

(42)

and (41) as

$$ \begin{aligned} {p_y}\left( {r,\theta } \right) =& \int\limits_{k \ge 0} {\int\limits_\varphi} {p_n}\left( {\sqrt {{k^2} + {r^2} - 2kr\cos \left( \varphi \right)} ,\varphi } \right)\\ \times& {p_s}\left( {k,\varphi } \right)kdkd\varphi. \end{aligned} $$

(43)

The PDF of the background noise is given by

$$ {p_n}\left( {r,\theta } \right) = \frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}, $$

(44)

and the PDF of the received signal is obtained by substituting (44) in (43), and is given by

$$ \begin{aligned} {p_y}\left( r,\theta \right) =& \frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}\int\limits_{k \ge 0} {\int\limits_\varphi {{e^{ - \frac{{{k^2} - 2rk\cos \left( \varphi \right)}}{{\sigma _n^2}}}}{p_s}\left( {k,\varphi } \right)kdkd\varphi } } \\ =&\frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}\int\limits_{k \ge 0} {\int\limits_\varphi {{e^{ - \frac{{{k^2} - 2rk\cos \left( \varphi \right)}}{{\sigma _n^2}}}}\frac{{{p_{\left| s \right|}}\left( k \right)}}{2\pi}dkd\varphi } } \\ =& \frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}\int\limits_{k \ge 0} {{e^{ - \frac{{{k^2}}}{{\sigma _n^2}}}}{I_0}\left( {\frac{{2rk}}{{\sigma _n^2}}} \right){p_{\left| s \right|}}\left( k \right)dk} \\ =& \frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}{E_{_{\left| s \right|}}}\left\{ {{e^{ - \frac{{{{\left| s \right|}^2}}}{{\sigma _n^2}}}}{I_0}\left( {\frac{{2r\left| s \right|}}{{\sigma _n^2}}} \right)} \right\}\\ =& \frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}{E_{_{\left| s \right|}}}\left\{ {f\left( {\left| s \right|} \right)} \right\}\\ =&\frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}{E_{\left| x \right|}}\left\{ {f\left( {G\left( {\left| x \right|} \right)} \right)} \right\} \\ =&\frac{1}{{\pi \sigma _n^2}}{e^{ - \frac{{{r^2}}}{{\sigma _n^2}}}}\int\limits_{k \ge 0} {{e^{ - \frac{{G{{\left( k \right)}^2}}}{{\sigma _n^2}}}}{I_0}\left( {\frac{{2rG\left( k \right)}}{{\sigma _n^2}}} \right){p_{\left| x \right|}}\left( k \right)dk}, \end{aligned} $$

(45)

where $f\left( {\left| s \right|} \right)={e^{ - \frac{{{\left| s \right|^2}}}{{\sigma _n^2}}}}{I_0}\left( {\frac{{2r\left| s \right|}}{{\sigma _n^2}}} \right),$ and the PDF of $\left|y\right|$ is given by

$$ p_{\left|y\right|}\left(r\right) = \frac{2r}{{\sigma_n^2}}{e^{-\frac{{{r^2}}}{{\sigma _n^2}}}}\int\limits_{k \ge 0} {{e^{-\frac{{G{{\left(k\right)}^2}}}{{\sigma_n^2}}}}{I_0}\left( {\frac{{2rG\left(k\right)}}{{\sigma _n^2}}} \right){p_{\left| x \right|}}\left( k \right)dk}. $$

(46)