Predistortion circuits can improve the linearity of power amplifiers (PAs) and other devices used in communications systems. An amplifying predistorter, for example, employs the nonlinear distortion components of a preamplifier to correct the distortion of a following nonlinear device, such as a PA, rather than using separate nonlinear elements such as diodes or field-effect transistors (FETs). Amplifying predistortion, which can be considered a generalization of feedforward (FF) technique, can be used to improve amplifier efficiency while reducing cost and size compared to traditional FF methods. The approach can be adapted to systems with frequency upconverters as well as to systems with broadband electro-optic devices.

FF systems date from the early 20th century,^{1,2} but their application in high-frequency systems began with the work of Seidel *et al.*^{3} Since then, numerous FF systems have been reported in the literature for modern applications.^{4-6} An example of amplifying predistortion (APD) can be found in ref. 7, where a driver amplifier is used to generate distortion to correct the distortion of the main PA. In this example, discussed in relationship to Class AB devices and operating over a narrow bandwidth, modest (9 dB) suppression of intermodulation distortion (IMD) was achieved.

APD is an attractive for several reasons. It does not suffer from the bandwidth restriction of a typical digital predistorter. In fact, it has been successfully applied to linearize devices spanning more than an octave. In addition, the APD approach has the following advantages over a conventional FF system:

- Losses due to the delay line and couplers at the output are eliminated.
- The error amplifier is of substantially lower power than used with a traditional FF system.
- Since delay elements are not at the output, they need not exhibit low loss; compact, low-cost delay lines can therefore be used.
- The method is amenable to adaptive linearization to correct for aging, drift, and temperature effects.
- It is possible to linearize devices with inputs having a different nature than the outputs. Examples include a millimeter-wave amplifier preceded by a frequency upconverter and electro-optical transducers such as lasers or electro-optic modulators.

For practical application of APD, it would be useful to estimate the amount of IMD suppression that can be achieved by this technique compared to high-performance FF techniques. By treating the APD approach as a generalized form of FF, it may be possible to arrive at some estimates. Figure 1(b) shows a basic block diagram for the APD scheme; it bears a superficial resemblance to the FF system shown in Fig. 1(a).

However, unlike an FF system where the distortion generated by a given device is canceled at its output, in an APD a second device (A3, the driver stage) is used as an additional source for distortion components. The distortion from A3 is fed with the correct magnitude and phase at the *input* of power stage A1 in order to cancel the distortion components generated by A1. If A1 is removed from the circuit, the distortion from the path consisting of A2 does not completely cancel the distortion from the main path as in a FF scheme. But the extra gain of A2 helps generate more distortion than necessary compared to a FF scheme, and this extra distortion cancels the distortion from the power stage (A1). For this reason, this technique can be called the over-compensated feedforward (OCFF) technique.

In order to develop a simplified OCFF analysis approach, fifth- and higher-order distortion components will be neglected, a reasonable assumption for Class A amplifiers and certain electro-optic components. It will also be assumed that the imaginary part of the devices' nonlinear transfer function is negligible.^{8,9}

Referring to Fig.1(b), the following parameters can be defined as:

P_{1} = the input power (two-tone excitation) at the input port of A3 (in dBm/tone);

IM3_{A1} and IM3_{A3} = the two-tone third-order intermodulation components at the outputs of A1 and A3, respectively (in dBm);

IP3_{A1} and IP3_{A3} = the output third-order intercepts of A1 and A3, respectively (in dBm);

G_{A1}, G_{A2}, and G_{A3} = the gain of A1, A2, and A3, respectively (in dB);

L = the loss of the main path, which is comprised of the insertion losses of couplers C1 and C2 and the loss through delay line D1 (in dB).

At point A of Fig.1(b), following the definition of third-order intercept point:

At point B, the intermodulation voltage signal from the main path is:

and the intermodulation voltage signal from the shunt path at point B is:

where:

G_{shunt} = the gain in the shunt path comprised of G_{A2} minus coupling losses in C1 and C2 and insertion loss in hybrid coupler H1.

If the delays in the main and shunt paths are identical, the phasors V_{1} and V_{2} would be in opposite phase, provided that gain G_{A2} exceeds some minimum value. In that case, the resultant voltage at point B can be expressed as:

At point C, the intermodulation voltage signal from A1 alone can be expressed as:

Referring the above signal to the point B results in:

which can be converted to voltage to obtain:

Cancellation of the distortion phasors requires that:

From Eqs. 2, 4, and 5, it is possible to solve for G_{shunt} to get:

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It can be seen that IP3_{A3} − L is the third-order intercept at point B from amplifier A1 and IP3_{A1} − G_{A1} is the third-order intercept from amplifier A1, also referenced to point B. If IP3_{A3} − L = IP3_{A1} − G_{A1}, then G_{shunt} = −L + 6 which can be interpreted as follows. The extra gain in the shunt path overcomes the loss L (as in FF) and further doubles the distortion signal to compensate the distortion from A1. For the hypothetical case of A1 being perfectly linear (IP3_{A1} going to infinity), the OCFF scheme degenerates to a traditional FF scheme, and G_{shint} becomes −L. Figure 2 graphically depicts shunt gain as a function of the driver-stage output third-order intercept point.

An estimate of the figure of merit of the OCFF scheme can be made based on the assumption that the distortion phasors are perfectly matched in magnitude but possesses a phase error, Θ, between them. Figure 3(a) shows the phasor diagram of a FF scheme, where the suppression of the distortion phasor with respect to the original distortion phasor is calculated from elementary geometry to be 20log. Figure 3(b) shows the phasor diagram for the OCFF scheme, with all phasors referred to point B of Fig. 1(b). The resultant phasor due to the distortion from A1 and the shunt path is at an angle Φ from the reference as shown. Assuming that the phase angle of the distortion phasor from A1 is zero, the suppression of the final distortion phasor with respect to the original distortion phasor is given by 20log. From simple geometry, it follows that:

Figure 4 shows the suppression as function of phase angle error Θ. It can be seen that OCFF is always inferior to FF as far as this figure of merit goes, approaching FF performance for infinitely large values of G_{shunt}. However, large G_{shunt} values also imply low levels of distortion from A1, thereby defeating the purpose of the OCFF approach. The amount of noise injected into the system also becomes large for high values of G_{shunt}. Nevertheless, the absence of loss at the output of A1 in the OCFF approach more than compensates for this reduction in suppression.

If the imaginary part of the active devices' nonlinear transfer function is negligible, distortion cancellation on the order of 20 dB is realistic. Amplifier A2 must be linear, but does not have to deliver high amounts of power compared to the traditional FF scheme since it is feeding the distortion components to the input of the power stage rather than at its output.

For Class AB operation, fifth- and higher-order IMDs must be considered. Modeling may be further complicated when the phases of the IMDs become dependent on the power level. Therefore, the problem reduces to matching the nonlinear transfer function of devices A1 and A3 in an appropriate manner to achieve optimum suppression of the IMDs (a topic for future discussion). It should be noted that the method mentioned for Class A operation is a special case of this generalized approach. Intuitively, OCFF operation will be optimized when A1 and A3 will operate with the same conduction angles.

Assuming output power of P_{2} in both cases, the efficiencies of the FF and OCFF techniques can be compared. The parameters of interest are pure numbers (not in units) and can be expressed as:

g_{A1}, g_{A2}, and g_{A3} = the gains

of amplifiers A1, A2, and A3,

respectively;

c1, c2,and c3 = the coupling coefficients of couplers C1, C2, and C3, respectively (less than 1); and

l_{m} = the loss in the main path, given by l_{m} = ( 1 − c1)l_{d}(1 − c2)

where:

l_{d} = the loss in the main path delay line.

Power dissipation in A3 plays a negligible role in computing the overall efficiency as long as gain g_{A1} is large enough. Therefore, only power dissipation in A1 and A2 will be considered. If η1 is the efficiency of amplifier A1, the DC power dissipation values by A1 for the FF and OCFF cases, respectively, are:

To calculate the DC power dissipation in amplifier A2, two distinct cases will be considered. In case 1, distortion components at the output of H1 are orders of magnitude smaller than the distortion components of the main carriers, which is usually true for Class A operation. Therefore, A2 essentially handles the power from the suppressed carriers, determined by the amount of cancellation available from H1. The RF power levels at the output of A2 for the FF and OCFF cases, respectively, are P_{2}sg_{A2}/l_{m} and P_{2}sg_{A2}/(g_{A1}l_{m}, where s is the suppression on the carrier obtained at the output of H1.

Consequently, the DC power dissipation amounts by A2 for the FF and OCFF cases, respectively, are:

It should be noted that A2 is most likely to be a Class A amplifier since it must handle signals with large peak-to-RMS ratio. Therefore, although the amplifier for the FF case must be more powerful than that for the OCFF case, both would have same efficiency, η_{2}.

Designating η_{FF} and η_{OCFF} as the overall efficiencies for the FF and OCFF approaches, respectively, it is possible to determine from Eqs. 8(a) and 8(b) and 9(a) and 9(b) that:

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Since l_{m}g_{A1 }> 1, and l_{m }< 1, the efficiency of an OCFF system is always larger than that of an FF system.

In case 2 for the evaluation of efficiency, the distortion components at the output of H1 are large compared to the main carriers, which is often true for Class AB operation. For an FF system, the RF power delivered at the output of A2 is given by P_{2}(δ/c2), where δ is the ratio of the distortion power to the carrier power before linearization. For an OCFF system, the RF power at the output of amplifier A2 is (P_{2}/g_{A1})(δ/c2)o_{f}, where o_{f} is the overcompensation factor (greater than 1) that accounts for the excess signal required as explained earlier.

Following similar arguments as in Case 1, the ratio of the efficiencies of OCFF and FF can be shown as:

For practical applications, g_{A1 }> o_{f} and as in case 1, the efficiency of an OCFF system is always larger than that of an FF system.

From the above analysis, the improvement in efficiency offered by the OCFF approach compared to the traditional FF approach is evident, where the effects of eliminating main line loss and using a smaller error amplifier (A2) were demonstrated. The OCFF can use a higher-loss and lower-cost delay line than the FF approach, such as a lumped-element delay line. The lower-power dissipation of A2 translates into lower cost and smaller size for the overall system.

Experiments performed with Class A amplifiers demonstrated the OCFF concept. The output third-order intercept of main amplifier A1 and driver amplifier were +35.5 and +30.5 dBm, respectively. The main path delay line exhibited loss of about 2.5 dB at 800 MHz and delay of about 10 ns. Results for both approaches are shown in Figs. 5(a) and 5(b). Without linearization, amplifier A1 produces third-order intermodulation distortion levels of −40 dBc in the 700-MHz band and −42 dBc in the 800-MHz band for output power of +15 dBm per tone. Using the OCFF linearizer, suppression levels for third-order intermodulation distortion were more than 18 and 15 dB in the 700- and 800-MHz bands, respectively.

The OCFF technique provide comparable linearization performance to the FF approach. The technique is especially attractive since it eliminates any loss at the output of the power device from delay line and coupler as in a FF scheme. It also improves the overall efficiency of the system compared to a FF scheme. The technique can be applied where the output is of a different nature from the input, as in electrical to optical conversion, or a transmitter stage involving frequency upconversion with linearization is performed at intermediate frequencies (IFs). The OCFF approach allows the linearization of direct modulated lasers where the distortion depends on the link distance due to the interaction of the laser chirp with fiber dispersion. There is no limitation on the bandwidth of the OCFF approach, and the technique can be implemented where both even- and odd-order distortion components must be controlled.

REFERENCES

- H.S. Black, "Translating system," US Patent 1.686,792, Issued 9 October 1928.
- H.S. Black, U.S. Patent 2,102,671, Issued December 1937.
- H. Seidel, "A Microwave Feedforward Experiment,"
*Bell System Technical Journal*, Nov. 1971, pp. 2879-2916. - Peter B. Kennington,
*High-Linearity RF Amplifier Design*, Artech House, Norwood, MA, 2000, pp. 251-348. - Nick Pothecary,
*Feedforward Linear Power Amplifiers*, Artech House, Norwood, MA, 1999. - Steve C. Cripps,
*Advanced Techniques in RF Power Amplifier Design*, Artech House, Norwood, MA, 2002, pp. 197-255 - J. Yi, Y. Yang, M. Park, W. Kang, and B. Kim., "Analog Predistortion Linearizer for High-Power RF Amplifiers,"
*IEEE Transactions on Microwave Theory & Techniques*, Vol. 48, pp. 2709-2713, Dec. 2000. - Steve Maas,
*Nonlinear Microwave Circuits*, Artech House, Norwood, MA, 1988. - S. Mukherjee, "Vector Measurement of 'Non-linear Transfer Function',"
*IEEE Transactions on Instrumentation and Measurement*, August 1994. - S. Mukherjee, S.Y. Yuan, and J. Rai, "Linearization of Electrical and Electro-optic Signals," US patent application 2001.