Triangular Distribution-Based Companding Technique for Reducing PAPR of OFDM Systems

Abstract

Orthogonal frequency division multiplexing (OFDM) technology promises to be a key technique for achieving high data rates in the present as well as future broadband wireless communication systems. In spite of several advantages, one of the major drawbacks of OFDM system is its high peak to average power ratio (PAPR), which degrades the overall performance and results in many restrictions for practical applications. To reduce the PAPR, a novel PAPR reduction technique is proposed here that provides optimum bit error rate (BER) performance, better design flexibility and more degrees of freedom compared to other existing companding schemes. In this proposed technique, original distribution function of OFDM signal magnitude is transformed into a triangular distribution function for achieving a favorable trade-off between PAPR reduction and BER performance. A theoretical performance study with general formulas of the proposed scheme is presented and a comparative analysis is done with the existing algorithms. Extensive simulation results demonstrate that our proposed scheme can substantially outperform the existing non-linear companding transform (NCT) techniques in terms of PAPR reduction, BER and power spectrum density (PSD) performance.

Appendix

To calculate inverse cumulative distribution function from (8)

$$\begin{aligned} F_{Y}^{\text{TR}} &= \frac{{\left( {y - a} \right)^{2} }}{{\left( {b - a} \right)\left( {c - a} \right)}},\;a < y \le c \hfill \\ {\text{Let}},\; Y &= \frac{{\left( {y - a} \right)^{2} }}{{\left( {b - a} \right)\left( {c - a} \right)}} \hfill \\ {\text{or}}, \;\left( {y - a} \right)^{2} &= \left( {b - a} \right)\left( {c - a} \right)Y \hfill \\ {\text{or}}, \left( {y - a} \right) &= \sqrt {\left( {b - a} \right)\left( {c - a} \right)Y} \hfill \\ {\text{or}}, y &= \sqrt {\left( {b - a} \right)\left( {c - a} \right)Y} + a \hfill \\ \therefore\, F_{Y}^{{{\text{TR}} - 1}} \left( y \right) &= Y^{ - 1} = \sqrt {\left( {b - a} \right)\left( {c - a} \right)y} + a \hfill \\ \end{aligned}$$

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Further, from Eq. (8) we get

$$\begin{aligned} F_{Y}^{\text{TR}} &= 1 - \frac{{\left( {b - y} \right)^{2} }}{{\left( {b - a} \right)\left( {b - c} \right)}} , c < y \le b \hfill \\ {\text{Let}}, \;Y& = 1 - \frac{{\left( {b - y} \right)^{2} }}{{\left( {b - a} \right)\left( {b - c} \right)}} \hfill \\ {\text{or}}, \;\frac{{\left( {b - y} \right)^{2} }}{{\left( {b - a} \right)\left( {b - c} \right)}} &= 1 - Y \hfill \\ {\text{or}}, \;\left( {b - y} \right)^{2} &= \left( {b - a} \right)\left( {b - c} \right)\left( {1 - Y} \right) \hfill \\ {\text{or}}, \;\left( {b - y} \right) &= \sqrt {\left( {b - a} \right)\left( {b - c} \right)\left( {1 - Y} \right)} \hfill \\ {\text{or}}, y &= b - \sqrt {\left( {b - a} \right)\left( {b - c} \right)\left( {1 - Y} \right)} \hfill \\ \therefore \,F_{Y}^{{{\text{TR}} - 1}} \left( y \right) &= Y^{ - 1} = b - \sqrt {\left( {b - a} \right)\left( {b - c} \right)\left( {1 - Y} \right)} \hfill \\ \end{aligned}$$

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The expanding function C ⁻¹_TR (r _n ) of our proposed scheme can be obtained using Eq. (12)

$$\begin{aligned} {\text{Let}}, Y &= \sqrt {\left( {b - a} \right)\left( {c - a} \right)\left( {1 - e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} } \right)} + a, \hfill \\ {\text{or}}, \;Y - a &= \sqrt {\left( {b - a} \right)\left( {c - a} \right)\left( {1 - e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} } \right)} \hfill \\ {\text{or}}, \;\left( {b - a} \right)\left( {c - a} \right)\left( {1 - e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} } \right) &= \left( {Y - a} \right)^{2} \hfill \\ {\text{or}},\; 1 - e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} &= \frac{{\left( {Y - a} \right)^{2} }}{{ \left( {b - a} \right)\left( {c - a} \right)}} \hfill \\ {\text{or}}, \;e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} &= 1 - \frac{{\left( {Y - a} \right)^{2} }}{{ \left( {b - a} \right)\left( {c - a} \right)}} \hfill \\ {\text{or}}, \; - \frac{{y^{2} }}{{\sigma^{2} }} &= ln\left| {1 - \frac{{\left( {Y - a} \right)^{2} }}{{ \left( {b - a} \right)\left( {c - a} \right)}}} \right| \hfill \\ {\text{or}},\;y^{2} &= \sigma^{2} ln\left| {\frac{{\left( {b - a} \right)\left( {c - a} \right)}}{{\left( {b - a} \right)\left( {c - a} \right) - \left( {Y - a} \right)^{2} }}} \right| \hfill \\ {\text{or}},\;y& = \sigma \sqrt {{ \ln }\left| {\frac{{\left( {b - a} \right)\left( {c - a} \right)}}{{\left( {b - a} \right)\left( {c - a} \right) - \left( {Y - a} \right)^{2} }}} \right|} \hfill \\ \therefore\, C_{\text{TR}}^{ - 1} \left( y \right) &= C_{\text{TR}}^{ - 1} \left( {r_{n} } \right) &= Y^{ - 1} \hfill \\ &= \sigma \sqrt {{ \ln }\left| {\frac{{\left( {b - a} \right)\left( {c - a} \right)}}{{\left( {b - a} \right)\left( {c - a} \right) - \left( {y - a} \right)^{2} }}} \right|} \hfill \\ \end{aligned}$$

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Further using Eq. (12) we obtain

$$\begin{aligned} {\text{Let}},\;Y &= b - \sqrt {\left( {b - a} \right)\left( {b - c} \right)e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} } \hfill \\ {\text{or}}, \; \left( {b - a} \right)\left( {b - c} \right)e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} &= \left( {b - Y} \right)^{2} \hfill \\ {\text{or}},\;e^{{\left( { - \frac{{y^{2} }}{{\sigma^{2} }}} \right)}} &= \frac{{\left( {b - Y} \right)^{2} }}{{\left( {b - a} \right)\left( {b - c} \right)}} \hfill \\ {\text{or}}, - \frac{{y^{2} }}{{\sigma^{2} }} &= { \ln }\left| {\frac{{\left( {b - Y} \right)^{2} }}{{\left( {b - a} \right)\left( {b - c} \right)}}} \right| \hfill \\ {\text{or}}, y &= \sigma \sqrt {{ \ln }\left| {\frac{{\left( {b - a} \right)\left( {b - c} \right)}}{{\left( {b - Y} \right)^{2} }}} \right|} \hfill \\ \therefore \,C_{\text{TR}}^{ - 1} \left( y \right) &= C_{\text{TR}}^{ - 1} \left( {r_{n} } \right) = Y^{ - 1} = \sigma \sqrt {{ \ln }\left| {\frac{{\left( {b - a} \right)\left( {b - c} \right)}}{{\left( {b - y} \right)^{2} }}} \right|} \hfill \\ \end{aligned}$$

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