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Complexity-reduced Volterra series model for power amplifier digital predistortion

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Abstract

This paper expounds a complexity-reduced Volterra series model for radio frequency power amplifier (PA) behavioral modeling and digital predistortion (DPD). An analysis was conducted, which took into account the memory effect mechanisms of the PA. This led to a closed-form expression that relates the memoryless behavior of the PA to the finite impulse response feedback filter, which approximates the memory effects’ behavior. The analysis resulted in a complexity-reduced Volterra series model which allows for a substantial reduction in the requirements for digital signal processors and the time needed to construct and implement the DPD in a real-time environment. The proposed model was validated as a behavioral model and a DPD using two different PA architectures, employing two different transistor technologies, driven by both 20 MHz 1001 wideband code division multiple access and long term evolution signals. The results obtained demonstrate the excellent modeling and linearization capability of the complexity-reduced Volterra series model.

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Acknowledgments

This work was supported by the University of Waterloo, Waterloo, ON, Canada, and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like also to acknowledge the support of Agilent Technologies, for their donation of Advanced Design System software, and Freescale Semiconductor, for their donation of power amplifiers.

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Authors and Affiliations

  1. Emerging Radio Systems Research Group, Department of Electrical and Computer Engineering, University of Waterloo, 200 University Avenue West, Waterloo, N2L 3G1, ON, Canada

    Farouk Mkadem & Slim Boumaiza

  2. University of British Columbia, Vancouver, BC, Canada

    Marie Claude Fares

  3. Maxim Labs, Maxim Integrated Products, Inc., Sunnyvale, CA, USA

    John Wood

Authors
  1. Farouk Mkadem
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  2. Marie Claude Fares
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  3. Slim Boumaiza
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  4. John Wood
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Corresponding author

Correspondence to Farouk Mkadem.

Appendix

Appendix

$$\begin{aligned} y\left( n \right) &= G\left( {x\left( n \right) + \sum\limits_{j = 1}^{M} {\gamma_{j} y\left( {n - j} \right)} } \right)\mathop \simeq \limits_{{y\left( {n - j} \right) \simeq G\left( {x\left( {n - j} \right)} \right)}} G\left( {x\left( n \right) + \sum\limits_{j = 1}^{M} {\gamma_{j} G\left( {x\left( {n - j} \right)} \right)} } \right) = \sum\limits_{i = 1}^{N} {p_{i} \left( {x\left( n \right) + \sum\limits_{j = 1}^{M} {\gamma_{j} G\left( {x\left( {n - j} \right)} \right)} } \right)^{i} } \\ &= \sum\limits_{i = 1}^{N} {\sum\limits_{k = 0}^{i} {\left\{ {p_{i} \left( {\begin{array}{*{20}c} i \\ k \\ \end{array} } \right)x^{i - k} \left( n \right)\left( {\sum\limits_{j = 1}^{M} {\gamma_{j} G\left( {x\left( {n - j} \right)} \right)} } \right)^{k} } \right\}} } \\ &= \sum\limits_{i = 1}^{N} {p_{i} x^{i} \left( n \right)} + \left( {\sum\limits_{i = 1}^{N} {p_{i} \left( {\begin{array}{*{20}c} i \\ 1 \\ \end{array} } \right)x^{i - 1} \left( n \right)} } \right)\left( {\sum\limits_{j = 1}^{M} {\gamma_{j} G\left( {x\left( {n - j} \right)} \right)} } \right) + \sum\limits_{i = 1}^{N} {\sum\limits_{k = 2}^{i} {\left\{ {p_{i} \left( {\begin{array}{*{20}c} i \\ k \\ \end{array} } \right)x^{i - k} \left( n \right)\left( {\sum\limits_{j = 1}^{M} {\gamma_{j} G\left( {x\left( {n - j} \right)} \right)} } \right)^{k} } \right\}} } \\ &= G\left( {x(n)} \right) + \left( {\sum\limits_{i = 1}^{N} {ip_{i} x^{i - 1} \left( n \right)} } \right)\left( {\sum\limits_{j = 1}^{M} {\gamma_{j} G\left( {x\left( {n - j} \right)} \right)} } \right) + \sum\limits_{i = 1}^{N} {\sum\limits_{k = 2}^{i} {p_{i} \left( {\begin{array}{*{20}c} i \\ k \\ \end{array} } \right)x^{i - k} \left( n \right)\left( {\sum\limits_{j = 1}^{M} {\gamma_{j} G\left( {x\left( {n - j} \right)} \right)} } \right)^{k} } } \\ &= G\left( {x(n)} \right) + \left( {\sum\limits_{i = 1}^{N} {ip_{i} x^{i - 1} \left( n \right)} } \right)\left( {\sum\limits_{j = 1}^{M} {\gamma_{j} \left( {\sum\limits_{k = 1}^{N} {p_{k} x^{k} \left( {n - j} \right)} } \right)} } \right) + {\text{terms including higher order of }}\gamma_{j} \left( {i.e. \, \gamma_{j}^{a} ;a > 1} \right) \\ &\simeq G\left( {x(n)} \right) + \left( {\sum\limits_{i = 1}^{N} {ip_{i} x^{i - 1} \left( n \right)} } \right)\left( {\sum\limits_{j = 1}^{M} {\gamma_{j} \left( {\sum\limits_{k = 1}^{N} {p_{k} x^{k} \left( {n - j} \right)} } \right)} } \right) \\ &= G\left( {x(n)} \right) + \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{N} {\left\{ {ip_{i} p_{k} \gamma_{j} x^{i - 1} \left( x \right)x^{k} \left( {n - j} \right)} \right\}} } } = G\left( {x(n)} \right) + \sum\limits_{i = 0}^{N - 1} {\sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{N} {\left\{ {\left( {i + 1} \right)p_{i + 1} p_{k} \gamma_{j} x^{i} \left( x \right)x^{k} \left( {n - j} \right)} \right\}} } } \\ y\left( n \right) &= G\left( {x(n)} \right) + \sum\limits_{i = 0}^{N - 1} {\sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{N} {a_{ijk} x^{i} \left( x \right)x^{k} \left( {n - j} \right)} } } \\ \end{aligned}$$

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Mkadem, F., Fares, M.C., Boumaiza, S. et al. Complexity-reduced Volterra series model for power amplifier digital predistortion. Analog Integr Circ Sig Process 79, 331–343 (2014). https://doi.org/10.1007/s10470-014-0266-4

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