Techniques for Including Memory Effects in the Digital Pre-Distortion Process
By Murthy Upmaka, Agilent Technologies
Digital pre-distortion (DPD) is today the most popular technique for linearizing a power amplifier (PA). It linearizes the PA by using the real-world signal, which is complex and time varying, created in the base band using sophisticated signal processing. Modifying the signal to counter the nonlinear effects of the PA is done using additional signal processing. The typical DPD process steps are shown Figure 1.
One of the important and critical features of any PA is memory. The instantaneous output from the PA may depend not only on current samples, but also on past samples. Consequently, the DPD algorithm should be able to account for this behavior, as well as the nonlinearity. The input and output waveforms can be obtained from either a laboratory measurement of the actual PA hardware (HW) or from simulation of a PA model. If the DPD is performed on PA HW then the true behavior of the PA is captured. If the DPD is done in simulation then two approaches are followed:
1. A model of the PA must first be created and then the DPD process is applied
2. The actual circuit of the PA in the analog world is co-simulated with the DPD algorithm in the digital world
Understanding the treatment of memory effects in each of these implementations is critical to achieving the most efficient and linear PA output performance. Typically, the memory of a PA is interpreted from the intermodulation levels at the output or by plotting the AM-AM and AM-PM curves. Let’s take a closer look.
DPD on a PA Behavioral Model using Measurement-Based Polynomial Fitting
A polynomial-based PA model is curve fitted to the measured parameters such as gain, frequency, 1-dB compression point, 3rd order intercept point, etc. While this approach describes harmonic nonlinearities quite well, it does not fully account for certain RF impairments (e.g., the large signal input and output termination mismatch effects at the fundamental and harmonic frequencies). More importantly, the PA’s memory effects cannot be captured using the polynomial fit model. Other more complex and advanced PA modeling techniques exist (e.g., the Voltera series, Hammerstein polynomials, Weiner-Hamerstein polynomials, and nonlinear complex envelope impulse response models), which can potentially account for PA memory effects. However, when well fitted, these advanced models can result in highly complicated polynomials with an excessively large number of terms. As a result, simulation speed and robust convergence may be negatively impacted.
DPD on a PA Behavioral Model using Measurement-Based X-Parameters*
The X-parameter-based PA model offers an accurate, modern, stable, transportable, and robustly convergent nonlinear measurement-based representation of the PA’s nonlinear behavior. Since X-parameters are fully measurement based, they are highly accurate, especially in regards to frequency, power, harmonics, compression, bias dependencies, and input/output large signal complex impedance matching. To cover the scope of the model’s measurement needs in multiple dimensions (e.g., frequencies, power levels, bias supplies, and input and output load conditions), large amounts of data must be collected. Also, accuracy greatly depends on the number of harmonics covered in measurement. Theory to include the dynamic aspects of the memory is available in literature.  In the current implementation of X-parameter measurement-based models, longer term dynamic memory effects cannot yet be accounted for. However, there has been an adaption to account for static memory effects due to PA bias modulation. 
D on a Simulation-Based Behavioral Model
Using the Envelope Transient solver available in the Agilent EEsof GoldenGate RF simulation environment (operating from within the Cadence Virtuoso environment), one can accurately extract a high fidelity PA model that accounts for nonlinearities and memory effects.  These GoldenGate simulation-based extractions are called Fast Circuit Envelope (FCE) models. Most commonly applied to Radio Frequency Integrated Circuit (RFIC) designs, the designer configures multiple simulations in the dimensions of interest, such as power, frequency, bias, source, and load impedances, or any other parameter that can be controlled in the RFIC fabrication or measurement process. This method presents a viable, highly accurate, very quick and robust design and analysis flow for integrating the FCE simulation-based behavioral model within the DPD process. However, the designer could be required to perform a large (one time) number of simulations to capture the required behavior of the PA over the various dimensionalities of interest. If during the DPD process it is found that the scope of the FCE model has to be widened, then one has to repeat the entire process of model extraction.
DPD with Direct Native Circuit-Level Co-Simulation
Co-simulation with a native RF analog circuit-based model, described by an RF analog circuit simulator such as Agilent EEsof’s Advanced Design System, represents the optimum method for the DPD architect to capture the complete memory effects in the process of DPD amplifier linearization. Several other popular techniques to co-simulate DSP and RF circuits rely on extracting a baseband equivalent model of the RF circuit and then performing the DPD signal processing. These methods may completely ignore any PA memory effects, essentially forcing the designer to once again face many of the limitations previously explained.
There is one technique where no compromises are necessary and all the physics from the RF circuitry is captured accurately. However, a co-simulation of this nature between analog and digital worlds may be very long and potentially impractical.
To better understand the limitations, consider the following example of simulation analysis steps required using a traditional SPICE (time domain) simulator:
• Frequency range of simulation: 1800 MHz RF carrier
• Significant 3rd harmonic content (at 5400 MHz)
• 24.3-kHz symbol rate: simulation over 512 symbols
• 10 samples/period of highest tone of interest
• Simulation analysis steps = (5400e6 * 10) * (512 / 24.3e3) = 1.14e9. This is huge!
This could take quite a long time, depending on the complexity of the PA topology.
Fortunately, there is a solver technology called Circuit Envelope (CE), available within Agilent EEsof’s ADS, for RF and microwave engineers.  CE is a hybrid solution technique incorporating the best features and benefits from each of the Harmonic Balance (HB) and SPICE solvers. It is ideally suited for time varying (non-steady state) nonlinear analysis of circuits having both a high frequency (carrier) content and a lower frequency time variation (modulation), such as analog and digitally modulated complex format signals, including wireless standards compliant waveforms.
As shown in the simulation example, the major advantage of CE is the efficiency (speed increase) achievable. The bandwidth (BW) in the example signal was 24.3 kHz. At the third harmonic it is 3*24.3 = 72.9 kHz. Using CE, the solver only needs to sample the 72.9 kHz desired modulation BW, not the carrier’s third harmonic (at 5400 MHz). Again, assuming 10 samples per period, and collecting 512 symbols, we can determine the number of required simulation analysis points.
In this case, simulation analysis points = (3*24.3e3*10)*(512/24.3e3) = 15,360. This is significantly less than the 1.14e9 simulation analysis points required by a traditional SPICE (time domain) simulator. For this example, CE provides an efficiency factor greater than 74,000 times fewer simulation points!
DPAs implemented within Agilent EEsof’s ADS, the CE solver has a special option to extract the nonlinear PA model such that it includes only the instantaneous response, neglecting any history and memory effects. This option is called Automatic Verification Modeling (AVM), or “fast co-sim.” The advantage of utilizing the AVM mode is a significant increase in simulation speed, allowing for much quicker study/analysis of first-order nonlinear PA effects. Figures 2 and 3 show the AM-AM curves with and without AVM.
While the AVM option neglects memory effects, the DPD architect can take advantage of this feature to examine the efficacy of the DPD algorithm if there were no memory effects. This can establish a healthy exchange between the PA designer and the DPD algorithm architect, as the PA designer would know the extent of memory reduction required to achieve better linearization.
DPD on PA HW Using Measurement Instruments
As already mentioned, when the DPD algorithm is tested on PA HW, all the memory effects can be captured. However, the measurement instruments and associated HW (e.g., cables and adapters) must be carefully chosen because the imperfections in the HW can appear like memory. This can be illustrated with a simple simulation. For this case, assume that the output from a PA was captured with the AVM option turned ON, which produced a single valued AM-AM curve as in Figure 2. Then, imperfections were artificially created to imitate the actual HW. A 3 dB drooping frequency response in the pass band of the signal was created using a filter cascaded with the PA. The AM-AM curve in Figure 4 shows spreading that appears to be caused by memory effects, even though they are not.
The optimal means of fully accounting for the nonlinear behavior of an RF PA, including the memory effects in the DPD process, is to perform the DPD on a hardware amplifier in the lab using measurement instruments. Imperfections from the measurement system can manifest as memory effects, so care must be taken to isolate measurement effects from those of the PA. Building a DPD algorithm by co-simulating with a circuit level amplifier will deliver the most flexibility and accuracy. The DPD architect will understand the margins and limits of the DPD algorithm, while the RF PA engineer will understand how the PA design margins and limits can be improved or modified to achieve more efficient and linear PA output performance.
*X-parameters is a trademark of Agilent Technologies, Inc. The X-parameters format and underlying equations are open and documented. For more information on the use of this trademark, refer to X-parameters Open Documentation, Trademark Usage & Partnerships.
About the Author
Murthy Upmaka has been with HP and Agilent Technologies for the past 17 years as senior consultant, system architect and senior applications engineer in both R&D and in the field. He supports simulation and modeling of high frequency RF circuits and systems for both wireless communications and the aerospace/defense sectors.
J. Verspecht, J. Horn, L. Betts, Ch. Gillease, D. Gunyan and D. Root,”Extension of X-parameters to Include Long-Term Dynamic Memory Effects,” 2009 IEEE MTT-S Int. Microwave Symp. Dig., Boston, MA, USA, June 2009.
Memory effects in Microwave components: using X-parameters to characterize and model long term memory effects of wideband modulated signals, Application note 5990-5799EN, Agilent Technologies.
HS Yap, “Designing to digital wireless specifications using circuit envelope simulation”, Asia Pacific Microwave Conference Proceedings, 1997, p 173-176.
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