Spectral analysis of GFDM modulated signal under nonlinear behavior of power amplifier

Abstract

Generalized frequency division multiplexing (GFDM) is a flexible non-orthogonal waveform candidate for 5G which can offer some advantages such as low out-of-band emission and high spectral efficiency. This paper investigates the effects of nonlinear behavior of practical power amplifier (PA) on the GFDM spectrum. A closed form expression for power spectral density (PSD) of GFDM signal is extracted. Then, PSD at the output of PA as a function of input power and the coefficients of nonlinear polynomial PA model is derived. In addition, the adjacent channel power (ACP) and ACP ratio, as two important performance metrics, are evaluated. The simulation results confirm the accuracy of derived analytical expressions. Moreover, to validate the performance of GFDM modulation after nonlinear PA, it is compared with orthogonal frequency division multiplexing modulation.

Appendices

Appendix 1

Moments of complex Gaussian random variable l are given by [34]

$$\begin{aligned} \begin{aligned}&E[{{l}_{1}}{{l}_{2}}\ldots ,{{l}_{s}}l_{1}^{*}l_{2}^{*}\ldots ,l_{m}^{*}] \\&=\left\{ \begin{aligned}&0,s\ne m \\&\sum \limits _{\pi }{E[{{l}_{\pi (1)}}l_{1}^{*}]E[{{l}_{\pi (2)}}l_{2}^{*}],\ldots ,E[{{l}_{\pi (s)}}l_{m}^{*}]},s=m \\ \end{aligned} \right. \\ \end{aligned} \end{aligned}$$

(27)

where $\left\{ {{l}_{i}}, i=1,2,\ldots s,\ldots ,m \right\}$ are complex Gaussian random variables and $\pi$ is apermutation of the set of integers $\left\{ 1,2,\ldots ,s,\ldots ,m \right\}$ [29]. Due to Gaussian distribution of y(t), (27) is used to calculate ${{\phi }_{{{i}_{1}},{{i}_{2}}}}(t,\tau )=E[y{{(t)}^{{{i}_{1}}+1}}y{{(t-\tau )}^{{{i}_{2}}}}{{({{y}^{*}}(t))}^{{{i}_{1}}}}{{({{y}^{*}}(t-\tau ))}^{{{i}_{2}}+1}}]$ as

$$\begin{aligned} {{\phi }_{{{i}_{1}},{{i}_{2}}}}(t,\tau )=\sum \limits _{\pi }{E[{{y}_{\pi (1)}}{{y}^{*}}(t)],\ldots ,E[{{y}_{\pi ({{i}_{1}})}}{{y}^{*}}(t)]}E[{{y}_{\pi ({{i}_{1}}+1)}}{{y}^{*}}(t-\tau )],\ldots ,E[{{y}_{\pi ({{i}_{1}}+{{i}_{2}}+1)}}{{y}^{*}}(t-\tau )] \end{aligned}$$

(28)

where $s=m={{i}_{1}}+{{i}_{2}}+1$ and ${{y}_{i}}=\left\{ \begin{aligned}&y(t)\quad \quad \quad i=1,\ldots ,{{i}_{1}}+1 \\&y(t-\tau )\quad i={{i}_{1}}+2,\ldots ,{{i}_{1}}+{{i}_{2}}+1 \\ \end{aligned} \right.$

By doing some manual calculation on (28), ${{\phi }_{{{i}_{1}},{{i}_{2}}}}(t,\tau )$ is derived as

$$\begin{aligned} \begin{aligned} {{\phi }_{{{i}_{1}},{{i}_{2}}}}(t,\tau )=&\sum \limits _{p=0}^{\min ({{i}_{1}},{{i}_{2}})}{\left( \begin{aligned}&{{i}_{2}}+1 \\&p+1 \\ \end{aligned} \right) \left( \begin{aligned}&{{i}_{1}}+1 \\&p+1 \\ \end{aligned} \right) \left( \begin{aligned}&{{i}_{2}} \\&p \\ \end{aligned} \right) \left( \begin{aligned}&{{i}_{1}} \\&p \\ \end{aligned} \right) (p+1)!(p)!({{i}_{2}}-p)!({{i}_{1}}-p)!} \\&{{({{R}_{yy}}(t,\tau ))}^{p+1}}{{({{R}_{yy}}^{*}(t,\tau ))}^{p}} {{({{R}_{yy}}(t,0))}^{{{i}_{1}}+{{i}_{2}}-2p}} \\ \end{aligned} \end{aligned}$$

(29)

Appendix 2

By considering (8), (15) and (16), ${{\phi }_{{{i}_{1}},{{i}_{2}}}}(\tau )$ is derived. In the following, by taking FT of it, ${{\phi }_{{{i}_{1}},{{i}_{2}}}}(f)=\int \limits _{-\infty }^{\infty }{{{\phi }_{{{i}_{1}},{{i}_{2}}}}(\tau ){{e}^{-j2\pi f\tau }}}d\tau$ is calculated as

$$\begin{aligned} \begin{aligned}&{{\phi }_{{{i}_{1}},{{i}_{2}}}}(f)=\sum \limits _{p=0}^{\min ({{i}_{1}}, {{i}_{2}})}{{{T}_{{{i}_{1}},{{i}_{2}}}}} \\&\times \left( \begin{aligned}&\sum \limits _{\upsilon =-\infty }^{\infty } {\sum \limits _{{{m}_{1}},\ldots ,{{m}_{p+1}}=0}^{M-1} \quad {\sum \limits _{m_{1}^{\prime },\ldots ,m_{p}^{\prime }=0}^{M-1}\quad {\sum \limits _{m_{1}^{\prime \prime }, \ldots ,m_{{{i}_{1}}+{{i}_{2}}-2p}^{\prime \prime }=0}^{M-1}{{}}}}} \\&\left( \int \limits _{0}^{M{{T}_{s}}}{\sum \limits _{\upsilon =-\infty }^{\infty }{(\prod \limits _{j=1}^{p+1}{{{g}_{{{m}_{j}}}}(t-\upsilon {{T}_{B}})}\prod \limits _{{{j}^{\prime }}=1}^{p}{g_{m_{{{j}^{\prime }}}^{\prime }}^{*}(t-\upsilon {{T}_{B}})}{{\prod \limits _{{{j}^{\prime \prime }}=1}^{{{i}_{1}}+{{i}_{2}}-2p}{\left| {{g}_{m_{{{j}^{\prime \prime }}}^{\prime \prime }}}(t-\upsilon {{T}_{B}}) \right| }}^{2}})}}dt \right) \\&\times \left( \int \limits _{-\infty }^{\infty } {\prod \limits _{j=1}^{p+1}{g_{{{m}_{j}}}^{*}(t-\tau -\upsilon {{T}_{B}})} \prod \limits _{{{j}^{\prime }}=1}^{p}{{{g}_{m_{{{j}^{\prime }}}^{\prime }}}(t-\tau -\upsilon {{T}_{B}})}{{e}^{-j2\pi f\tau }}}d\tau \right) \\ \end{aligned} \right) \\&\otimes \left( (\sum \limits _{{{k}_{1}},\ldots ,{{k}_{p+1}}=0}^{K-1} \quad {\sum \limits _{k_{1}^{\prime },\ldots ,k_{p}^{\prime }=0}^{K-1}{\delta (f-\dfrac{(\sum \limits _{j=1}^{p+1}{{{k}_{j}}}- \sum \limits _{{{j}^{\prime }}=1}^{p}{{{k}_{{{j}^{\prime }}}}-\dfrac{K-1}{2}})}{{{T}_{s}}})}} \right) \\ \end{aligned} \end{aligned}$$

(30)

where

$$\begin{aligned} {{T }_{{{i}_{1}},{{i}_{2}}}}=\left( \begin{aligned}&{{i}_{2}}+1 \\&p+1 \\ \end{aligned} \right) \left( \begin{aligned}&{{i}_{1}}+1 \\&p+1 \\ \end{aligned} \right) \left( \begin{aligned}&{{i}_{2}} \\&p \\ \end{aligned} \right) \left( \begin{aligned}&{{i}_{1}} \\&p \\ \end{aligned} \right) (p+1)!(p)!({{i}_{2}}-p) !({{i}_{1}}-p)!{{(\alpha {{\overline{p}}_{x}})}^{{{i}_{1}}+{{i}_{2}}+1}}{{(K)}^{{{i}_{1}} +{{i}_{2}}-2p}}\frac{1}{M{{T}_{s}}}. \end{aligned}$$

(31)

By defining ${{\tau }^{\prime }}=t-\tau -\upsilon {{T}_{B}}$ and using linear convolution formula, (30) can be expressed as

$$\begin{aligned} \begin{aligned}&{{\phi }_{{{i}_{1}},{{i}_{2}}}}(f)= \sum \limits _{p=0}^{\min ({{i}_{1}},{{i}_{2}})}{{{T}_{{{i}_{1}},{{i}_{2}}}}} \\&\times \left[ \begin{aligned}&\sum \limits _{{{m}_{1}},\ldots ,{{m}_{p+1}}=0}^{M-1}\quad {\sum \limits _{m_{1}^{\prime },\ldots ,m_{p}^{\prime }=0}^{M-1}\quad {\sum \limits _{m_{1}^{\prime \prime },\ldots ,m_{{{i}_{1}}+{{i}_{2}}-2p}^{\prime \prime }=0}^{M-1}{B(f)}}} \\&\times \left( G_{_{{{m}_{1}}}}^{*}(f)\otimes \ldots \otimes G_{{{m}_{p+1}}}^{*}(f)\otimes {{G}_{m_{1}^{\prime }}}(-f)\otimes \ldots \otimes {{G}_{m_{p}^{\prime }}}(-f) \right) \\ \end{aligned} \right] \\&\otimes \left( (\sum \limits _{{{k}_{1}},\ldots ,{{k}_{p+1}}=0}^{K-1}\quad {\sum \limits _{k_{1}^{\prime },\ldots ,k_{p}^{\prime }=0}^{K-1}{\delta \left(f-\dfrac{\left(\sum \limits _{j=1}^{p+1}{{{k}_{j}}}-\sum \limits _{{{j}^{\prime }}=1}^{p}{{{k}_{{{j}^{\prime }}}}-\dfrac{K-1}{2}}\right)}{{{T}_{s}}}\right)}} \right) \\ \end{aligned} \end{aligned}$$

(32)

where B(f) is equal to

$$\begin{aligned} \begin{aligned} B(f)=&\int \limits _{0}^{M{{T}_{s}}}{\sum \limits _{\upsilon =-\infty }^{\infty }{(\prod \limits _{j=1}^{p+1}{{{g}_{{{m}_{j}}}}(t-\upsilon {{T}_{B}})}\prod \limits _{{{j}^{\prime }}=1}^{p}{g_{m_{{{j}^{\prime }}}^{\prime }}^{*}(t-\upsilon {{T}_{B}})}{{\prod \limits _{{{j}^{\prime \prime }}=1}^{{{i}_{1}}+{{i}_{2}}-2p}{\left| {{g}_{m_{{{j}^{\prime \prime }}}^{\prime \prime }}}(t-\upsilon {{T}_{B}}) \right| }}^{2}}{{e}^{-j2\pi f(t-\upsilon {{T}_{B}})}})}}dt \\&=\int \limits _{-\infty }^{\infty }{(\prod \limits _{j=1}^{p+1}{{{g}_{{{m}_{j}}}}(t)}\prod \limits _{{{j}^{\prime }}=1}^{p}{g_{m_{{{j}^{\prime }}}^{\prime }}^{*}(t)}{{\prod \limits _{{{j}^{\prime \prime }}=1}^{{{i}_{1}}+{{i}_{2}}-2p}{\left| {{g}_{m_{{{j}^{\prime \prime }}}^{\prime \prime }}}(t) \right| }}^{2}}{{e}^{-j2\pi f(t)}})}dt \\&=\left( \begin{aligned}&{{G}_{{{m}_{1}}}}(f)\otimes \ldots \otimes {{G}_{{{m}_{p+1}}}}(f)\otimes G_{m_{1}^{\prime }}^{*}(-f)\otimes \ldots \otimes G_{m_{p}^{\prime }}^{*}(-f)\otimes \\&G_{m_{1}^{\prime \prime }}^{*}(-f)\otimes \ldots \otimes G_{m_{{{i}_{1}}+{{i}_{2}}-2p}^{\prime \prime }}^{*}(-f)\otimes {{G}_{m_{1}^{\prime \prime }}}(f)\otimes \ldots \otimes {{G}_{m_{{{i}_{1}}+{{i}_{2}}-2p}^{\prime \prime }}}(f) \\ \end{aligned} \right) . \\ \end{aligned} \end{aligned}$$

(33)

By considering (32) and (33), ${{\phi }_{{{i}_{1}},{{i}_{2}}}}(f)$ is obtained.