The ultimate aim of studying power amplifier (PA) distortions is to design appropriate predistorters that will compensate for these distortions and ensure linear amplification of the signal to be transmitted. In Chapter 3, the similarities between behavioral modeling and digital predistortion (DPD) were briefly introduced. Various mathematical formulations that can be used to implement behavioral models as well as digital predistorters were thoroughly discussed in Chapters 4-7. Chapter 8 exposed the common steps of the behavioral modeling and DPD processes with a focus on the identification techniques employed for the synthesis of the model or predistorter function. In this chapter, the specificities of DPD are addressed. Although the analysis is carried out in this chapter for the case of a PA, the concepts and results still hold in the case where the transmitter’s analog front end is part of the device under test (DUT) to be linearized.

Conceptually, predistortion consists in implementing a nonlinear function upstream of the PA complementary to that of the amplifier to be linearized. Accordingly, the cascade made of the predistorter and the PA will operate as a linear amplification system as illustrated in the simplified block diagram of Figure 9.1. This figure also depicts sample amplitude modulation to amplitude modulation (AM/AM) and amplitude modulation to phase modulation (AM/PM) characteristics of the predistorter, the PA and the linearized power amplifier (LPA). The objective is to have a constant complex gain over the entire operating power range of the linearized amplifier. The power transfer characteristics of the predistorter, the PA and the LPA of Figure 9.1 are reported in Figure 9.2. This latter figure clearly illustrates that the predistorter is designed to generate a gain expansion that will compensate for the gain compression commonly observed in PAs. Since some class AB PAs as well as Doherty amplifiers tend to exhibit a gain expansion followed by a gain compression in their AM/AM characteristics, the predistorter has to compensate for these and thus must produce a gain compression followed by a gain expansion.

Behavioral Modeling and Predistortion of Wideband Wireless Transmitters, First Edition. Fadhel M. Ghannouchi, Oualid Hammi and Mohamed Helaoui.

Figure 9.1 Simplified block diagram of predistortion system and corresponding gain characteristics of each block

Figure 9.2 Power transfer characteristics involved in a predistortion system

Figure 9.3 Power transfer characteristics in predistortion systems

A rudimentary numerical example that illustrates the predistortion concept is given in Figure 9.3 in which the input and output powers as well as the gains of a nonlinear PA and its predistorter are presented. This figure also includes the input and output power levels as well as the gain of the resulting linearized PA. In this case, the PA is assumed to have a small signal gain of 20 dB and a saturation output power in the range of 23 dBm. Even though this example considers a memoryless PA that does not cause AM/PM distortions, the same concept can be extended to include phase distortions and memory effects. As shown in Figure 9.3, a gain compression is observed in the PA characteristic for input power levels of 1 dBm and above. To compensate for this, the predistortion function introduces a complementary gain expansion. It is important to note here that the maximum input power of the predistorter is 3 dBm while the maximum input power of the PA is 7 dBm. Typically, when the input power of the DPD exceeds 3 dBm, clipping will occur in order to avoid overdriving the PA. The saturation input power of the DPD is determined by the saturation output power, or equivalently, the saturation input power of the PA and the gain of the linearized PA. This aspect will be further discussed in the DPD normalization gain section.

To ensure linear amplification, for a given output power of the PA at instant n(P_{ou}t_PA(n)), the required power level at the input of the predistorter (Pm__{PD}(n)) can be determined by:

where G_{LPA} is the desired complex power gain of the linearized amplifier.

The output power of the predistorter (P_{out}__{PD}(n)) is simply the input power of the

^{PA (P}in_PA^{(n)):}

Thus, the input-output power characteristic of the predistorter can be easily obtained from that of the PA by normalizing the output gain of the amplifier using the desired linear gain and then swapping the input and output data as demonstrated by Equations 9.1 and 9.2.

Since the input and output powers of the amplifier are related through its instantaneous gain (G_{pa}(n)) according to:

then, the predistorter’s input power can be expressed as a function of the PA input power using:

Thus, it is possible to express the instantaneous gain of the predistorter from the measured characteristics of the PA by combining Equations 9.1-9.4. This leads to:

When AM/PM distortions are present, the phase distortions caused by the predistorter ( G_{PD}(n)) are:

| G_{lpa} and | G_{pa} (n) are the AM/PM distortions of the linearized PA and the PA, respectively.

In summary, once the AM/AM and AM/PM characteristics of the amplifier are measured, the AM/AM and AM/PM characteristics of the corresponding predistorter can be determined using Equations 9.4, 9.5, and 9.6. Having these desired predistorter characteristics, the models described in the previous chapters can be applied to accurately fit this dataset.