**Chokri Jebali, Ghalid Idir Abib, Eric Bergault, and Ali Gharsallah**

Developing a behavioral model for power amplifiers with memory effects requires including nonlinear phenomena. As shown in Part 1 (*Microwaves & RF*, June 2010, p. 84), nonlinear amplifier behavior is usually characterized by AM-AM and AM-PM conversion. As will be shown in Part 2, models incorporating these effects can be validated by the appropriate measurements.

Measurement results (* Fig. 5*) show significant improvement by preprocessing the input samples. The results also carry out the lower condition number of the orthogonal polynomial model, with the robustness of the pseudo-inverse calculation provided by the other models. The memory polynomial model used the same condition number as the conventional polynomial model in these comparisons.

In memory and memory-less model cases, the condition number of the orthogonal model is lower than the conventional polynomial model. It increases exponentially as a function of nonlinearity order K and memory depth Q, and becomes large even for moderate K and Q. In practice, the inversion of φ^{H} x φ can be more difficult than that of ?^{H} x ?. * Figure 5* shows the condition number for the matrix ?

^{H}x ? for 3000 samples. The condition number increases at a much slower rate as K and Q increase. A low condition number will ensure better numerical stability. Analysis of the importance of the condition number was performed in

**. 4 and 5 to show the orthogonal polynomial basis is advantageous to the envelope memory polynomial model and the conventional polynomial model.**

*Figs* The delay alignment between input and output data is a significant constraint. If not perfectly accounted for, it can cause additional dispersion in the modeled AM/AM and AM/PM. This delay is mainly due to the sampling rate of the input and output waveforms. An upsampling ratio is needed to accurately estimate the delay between the input and output data. For the test system (* Figs. 2 and 3*), the output sample depends only on input sample.

* Figure 6* shows delay alignment sensitivity for the input/ output QPSK signal under study. Measured and modeled data are not synchronous; the shift is caused by the delay between the input and output data.

*compare the polynomial model, envelope memory polynomial model, orthogonal polynomial model, and measurements after canceling delays between the PA's inputs and outputs.*

**Figures 7 and 8***shows the temporal waveform (output versus input samples) while*

**Figure 7***shows the Q versus I data in a QPSK constellations for modeled and measured data.*

**Fig. 8**Delay misalignments degrade the accuracy of any PA behavioral model, and can be considered a memory effect. To study this, a simulation sweep was first run on the DUT's nonlinearity order and memory depth for both the EMPM and OMPM models. The NMSEs between the model output and test results were calculated for each combination of model parameter. To examine the goodness of fit, 3000 input and output samples were used, with the PA output calculated according to Ref. 2 and NMSE calculated according to Ref. 11.

* Figure 9* shows the NMSE when the conventional and orthogonal basis functions are used. The general trend is that the NMSE first decreases with increasing polynomial nonlinearity order K and increasing polynomial memory depth Q. After a threshold of a nonlinearity order K, the NMSE is almost constant around -40 dB. When the delay tap Q increases with the same nonlinearity order as before, the NMSE decreases to a significant value compared to the memory-less conventional polynomial case (with Q = 2).

The robustness of the experimental test setup (* Figs. 2 and 3*) was evaluated with nonlinear 1575-MHz PA based on a Fujitsu model FLL107ME MESFET. The amplifier was tested with QPSK signals and a chip rate of 1 MSamples/s.

*shows the measured AM/PM compared with data from the conventional and orthogonal polynomial models. The AM/PM curves show some dispersion due to static nonlinearity and weak memory effects. Memory effects are attributed mainly to the nonconstant frequency response around the carrier frequency, the impedance variation of the bias circuits, and the harmonic loading of the power transistors. To minimize transistor nonlinearities, it was thought to use predistortion for linearization, applied by means of a load-pull/source-pull measurement system.*

**Fig. 10**^{16}

Memory effects can arise from narrowband signals, from thermal source, and from electrical sources. When a PA is driven by narrowband input signals, heating occurs at the device junction level. For wideband signals, memory effects are primarily electrical, due mainly to the frequency response of the bias circuit over the input modulation bandwidth. These electrical memory effects can be minimized by the design of the bias network, although they can still be present for signal bandwidths exceeding 10 MHz.

Because WCDMA signals occupy 5 MHz per carrier, thermal memory effects were not evident. Rather, for a DUT with multicarrier WCDMA signals, the memory effects are more electrical, appearing as dispersion.

In cases where the modulation varies amplitude, it would be logical to study the effects of input amplitude variations on output waveforms. The resulting distorted waveform can be analyzed for its frequency components. Phase distortion is usually characterized in terms of AM/ PM conversion. The most common manifestation of AMPM effects is an asymmetrical slewing of displays showing IMD or spectral regrowth. The cause of this effect is the key to quantifying AM-PM effects in individual cases.

Although power-series models can predict PA nonlinear behavior, they are limited because of their lack of a phase component in the output term. Power series coefficients are also sensitive to changes in input and output impedance and bias levels. These models tend to be effective for weakly nonlinear conditions. But PAs at or beyond compression can exhibit strongly nonlinear behavior. Because of the differences between strong and weak nonlinear effects, different nonlinear models are needed.