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.
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Appendices
Appendix 1
Moments of complex Gaussian random variable l are given by [34]
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
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
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
where
By defining \({{\tau }^{\prime }}=t-\tau -\upsilon {{T}_{B}}\) and using linear convolution formula, (30) can be expressed as
where B(f) is equal to
By considering (32) and (33), \({{\phi }_{{{i}_{1}},{{i}_{2}}}}(f)\) is obtained.
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Mohammadian, A., Mohammadi, A., Abdipour, A. et al. Spectral analysis of GFDM modulated signal under nonlinear behavior of power amplifier. Wireless Netw 27, 137–149 (2021). https://doi.org/10.1007/s11276-020-02403-2
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DOI: https://doi.org/10.1007/s11276-020-02403-2