TRANSFER FUNCTIONS
To evaluate the dependency of PA transfer functions AM/AM and AM/PM on the envelope frequency a simple two-tone test signal can be applied. Since the digital PD and OFDM system is supposed to be implemented at baseband frequency, we need to develop a discrete-time complex baseband PA model. Generally, the baseband modulated signal is fed to the digital to analogue converter (DAC) and IF/RF up- conversion as shown in the block diagram of Figure 2.15 (a). In order to represent the PA at baseband, the discrete time signal v[n] is considered as an oversampled version of the continuous-time signal S[t] [23].
Assuming the input bandpass signal to the PA as follows:
{
j ct}
(
)
c
41
Where g(t) is the complex envelope signal to the SSPA, is the carrier centre frequency and r(t) and θ(t) are the time-varying amplitude and phase of g(t), respectively. [ ] S t v t[ ] (a) [ ] V n (b)
Figure 2.15: (a) General RF transmitter and (b) baseband equivalent model
Accordingly, the equivalent baseband PA model for a bandpass memoryless nonlinearity can be described by polynomials as follows:
Where
Here are complex coefficients. By rewriting (2.19) using (2.17), the odd-order complex power series is defined as:
(2.18)
| |
42 In other words:
In which | | and are equivalent to AM/AM and AM/PM distortion respectively. By performing the two-tone test and using the complex envelope f(t), the PA transfer functions AM/AM and AM/PM can be derived. For this purpose a two single tone signal with magnitude of A/2 and phase of , with tone spacing equal to , is considered as:
/2 cos cos (2.22)
This can be simplified as follows:
cos . cos (2.23)
Observing the input signal envelope, it is found out the amplitude of input complex envelope signal, r(t), is cos . Thus, the output complex envelope f(t) can then be concluded as follows [19], [24]:
(2.24) (2.20)
43 2 1 (2.25) where 1 4 2 1 (2.26)
From equation (2.25), it is obvious that the transfer functions of PA depend on tone spacing, ωm, for two-tone signals [19]. To derive the frequency dependent coefficient
two-tone measurements for different values of tone-spacing and input amplitude can be performed [19]. Thus, the frequency-dependent complex power series considering memory effects can be described as:
, … (2.27)
(2.28)
Accordingly, the amplified signal at the out of PA considering memory effects is:
| , | cos ∠ , (2.29)
The goal is representing a black-box model addressing the memory effect and nonlinearity of PA by which the frequency-dependent complex polynomial in equation (2.28) can be realized.
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2.8 REFERENCES
[1] P. B. Kenington, High-Linearity RF Amplifier Design, Norwood, MA: Artech House, 2002.
[2] S. C. Cripps, “Advanced Techniques in RF Power Amplifier Design“, Norwood, MA: Artech House, 2002.
[3] F. Giannini, G. Leuzzi, “Nonlinear Microwave Circuit Design”, England, WS: John Wiley & Sons Ltd, 2004.
[4] G. Gonzalez, Microwave transistor amplifiers: analysis and design. Upper Saddle River, NJ: Prentice-Hall, 1997.
[5] M. S. O'Droma, J. Portilla, E. Bertran, T. J. Donati, S. Brazil, M. Rupp, and R. Quay, “Linearization Issues in Microwave Amplifiers", Proc. European Gallium Arsenide and other Compound Semiconductors Application Symposium (GAAS'04-EUMW), Oct. 2004.
[6] X. Zhang, L. E. Larson, P. M. Asbeck, “Design of Linear RF Out-phasing Power amplifiers“, Norwood, MA: Artech House, 2003.
[7] F. H. Raab, P. Asbeck, S.Cripps, P. B. Kenington, Z. B. Popovic, N. Pothecary, J. F. Sevic, N. O. Sokal, "Power amplifiers and transmitters for RF and microwave", IEEE Transactions on Microwave Theory and Techniques, vol. 50, pp. 814-826, Mar. 2002.
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[9] J. Minkoff, “Intermodulation Noise in Solid-State Power Amplifiers for Wideband Signal Transmission”, Communication Satellite Systems Conference, 9th, p. 304- 313, 1982.
[10] A. M. Salman, Analysis, Modelling and Linearization of Nonlinearity and Memory Effects in Power Amplifiers Used for Microwave and Mobile Communications, PhD Thesis, Kassel University, Germany, March 2005.
[11] W. Bosch, and G. Gatti, “Measurement and Simulation of Memory Effects in Predistortion Linearizers", IEEE Transaction on Microwave Theory & Technique, vol. 37, no. 12, pp. 1885-1890, Dec. 1989.
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[13] S. Boumaiza, and F. M. Ghannouchi, “Thermal Memory Effects Modelling and Compensation in RF Power Amplifiers and Predistortion Linearizers", IEEE Transaction on Microwave Theory & Technique, vol. 51, no. 12, pp. 2427-2433, Dec. 2003.
[14] M. Franco, “Minimizing Power Amplifiers Memory Effects", Proc. IEEE MTT International Microwave Symposium. Workshop on Distortion Correction of High Power Amplifiers Using Digital Signal Processing, Dallas, Texas, June 2004.
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[16] P. Perugupalli, Y. Xu, K.Shenai, "Measurement of thermal and packaging limitations in LDMOSFET's for RFIC application", Proc. IEEE Instrum. Meas. Technol. Conf., vol. 1, 1998, pp. 160-164.
[17] J. H. Vuolevi, T. Rahkonen, "Third-order intermodulation distortion caused by thermal power feedback", Proc. Norchip'99, Oslo, Norway, pp. 121-125, 1999.
[18] H. Ku and J. S. Kenney, “Behavioural modelling of nonlinear RF power amplifiers considering memory effects,” IEEE Transaction on Microwave Theory & Technique, vol. 51, no. 12, pp. 2495–2504, Dec. 2003.
[19] H. Ku, M. D. McKinley, and J. S. Kenney, “Quantifying memory effects in RF power amplifiers,” IEEE Transaction on Microwave Theory & Technique, vol. 50, no. 12, pp. 2843–2849, Dec. 2002.
[20] J. C. Pedro, N. B. Cavalho, “Intermodulation Distortion in Microwave and Wireless Circuit”, Artech House microwave Library, 2003.
[21] N. Borges de Carvalho and J. Carlos Pedro, “A Comprehensive Explanation of Distortion Sideband Asymmetries,” IEEE Transaction on Microwave Theory & Technique, vol. 50, no. 9, pp. 2090–2101, Sep. 2002.
[22] N. Borges de Carvalho and J. C. Pedro, “Large and small signal IMD behaviour of microwave power amplifiers,” IEEE Transaction on Microwave Theory & Technique, vol. 47, pp. 2364–2374, Dec. 1999.
[23] A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing, 3rd edition,
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[24] S. Yi, S. Nam, S. Oh, and J. Han,” Prediction of a CDMA output spectrum based on intermodulation products of two-tone test,” IEEE Transaction on Microwave Theory & Technique, vol. 49, no. 5, pp. 938-946, May 2001.
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