Overview of Power Amplifier Linearization Based

8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

Overview of Power Amplifier Linearization Based on Predistorsion Techniques E. BERTRAN, P.L. GILABERT, G. MONTORO, J.BERENGUER Department of Signal Theory and Communications Universitat Politecnica de Catalunya EPSC-UPC, Avda. Canal Olimpic, 15, 08860-Castelldefels SPAIN [plgilabert,bertran,montoro,berenguer]@tsc.upc.edu , http://cmc.upc.edu Abstract: - An overview of digital predistortion techniques for microwave power amplifier linearization is here presented. Different predistortion approaches are considered, and both static and dynamic (memory effects) behavioral models are presented. Special focus is done to the most suitable solutions for facing the new challenge to linearize high-bandwidth amplifiers, as demands most of the new telecommunications standards and emergent technologies related with cognitive radio. Key-Words: - Microwave Power Amplifiers, Linearization, Digital Predistortion, Behavioral models.

linearizers (regarding power consumption issues), whereas digital predistorters reduce the problem of parameter sensitivity (and adjustment) which is the main pitfall in analog predistorters. In the event of high signal bandwidth (tens of MHz), a dynamic predistorter is recommendable, which has to be implemented in digital technology because the need of compensating the memory effects. Anyway, the predistorter has to operate at high velocities (microwave frequencies) with moderated computational effort (in order do not enlarge the sampling period with has to be keep stuck in the DSP or FPGA devices because the aforementioned mandatory functionalities which have to coexist with linearization ones. Besides, the dynamic nonlinear model of the power amplifier is not completely observable, and difficult to be accurately obtained in real-time conditions. So, options to face a dynamic predistorter (non-linear in nature) such as most of the developed in classical non-linear Control theory (space state based), are usually not efficient. Simplified behavioral models are preferable.

1 Introduction Together with polar transmitters, digitally predistorted (DPD) power amplifiers (PA) are of the most suitable solutions to face the novel challenge of designing multiband-multimode transmitters. The classical drawbacks associated to their digital nature have been already overcome because the prior must of DSP (Digital Signal Processor) or FPGA (Field Programable Gate Array) devices to cope with modern standards (802.11, 802.16, DVB-T, CDMA, UMTS, LTE...), which include mandatory functionalities such as source coding, scrambling, interleaving and other processing requirements that have to be implemented in the digital domain. So, the need of these devices is no longer due to the PA linearization subsystem. On the contrary, it is possible to take advantage of their presence for also supporting digital PA linearization functionalities without any significant increment of the transmitter development cost. Conceptually, predistortion (PD) is a linearization technique quite intuitive since it consists in preceding the PA with a predistorter in order to counteract the nonlinear characteristic of the PA. The objective of the predistorter is then to ideally reproduce the inverse PA nonlinear behavior and as a result, having linear amplification at the output of the PA [1]. Fig. 1 shows the basic principles of an open-loop predistorter. According to the type of power amplifier and the signal bandwidth, the predistorter may be memoryless or dynamic. The first may be implemented on both analog and digital alternatives, being the analog ones quite usual in satellite

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Fig. 1. Fundamentals of predistortion linearization

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nonlinear distortion in the feedback loop (e.g. introduced by up&down-converters) when considering adaptive predistortion. This nonlinear distortion does not have to be compensated but can mask the open loop linearity performance and produce unwanted nonlinear compensations. One example of RF predistortion using analogue devices is the so called Cubic Predistorter. The cubic RF predistorter is aimed at canceling third-order IMD by adding a properly corrected in amplitude and phase cubic component to the input signal. In the case of a band-pass system only third-order IMD products are usually reduced, improvements considering high order IMD are usually little rewarded. In a typical cubic predistorter the input signal is split into two paths. The lower path is formed by a cubic nonlinearity performing the nonlinear predistortion, a gain (variable attenuator) and phase (variable phaseshifter) controllers that ensure a correct match at the combiner and finally, a post-distortion amplifier that buffers and amplifies the resulting signal. The amplifier in this path is considered to be a smallsignal device that does not contribute to the overall nonlinear distortion. In order to ensure a perfect match in the recombination process of both split signals, a time-delay element is added, usually in the upper path. Finally the predistorted signal in the lower path is combined with the signal in the upper one and fed to the PA. The combined signal is composed by a replica of the original one plus a nonlinear component which is intended to compensate the cubic characteristic of the PA. Notice the criticism of the gain and phase adjustments to this cubic characteristic, imperfect adjustment represent an undesired increase in IMD. The nonlinearity in the predistorter is usually created making use of the nonlinear characteristics of diodes: single diode, antiparallel diode, varactor-diode [2]; and also FET transistors [3]. On the other hand, digital predistortion makes use of digital processing devices such as DSPs or FPGAs. Fig. 3 shows the simplified block diagram of a transmitter containing an adaptive digital predistortion module at BB. Two main approaches can be found in digital predistortion which are signal and data predistortion. The main difference between signal and data predistortion regards the position in the transmitter chain where the predistortion is carried out [4]. The predistorter module in data predistortion is situated before the pulse-shaping filter, while in signal predistortion the predistortion is carried out after the pulse-shaping filter. Hence, Fig. 3 shows an example of signal predistortion.

2 Predistortion approaches Several solutions have been developed to realize the predistorter, from digital baseband processing, which represents the main scope of this overview, to processing the signal directly at RF or IF by using nonlinear devices such as diodes. Besides, most of current predistortion solutions already introduce some kind of feedback mechanism in order to allow a more robust operation of the linearizer. These feedback mechanisms may range from classical feedback, which allows compensating small parameter variations, to adaptive structures. Therefore all different solutions proposed for the realization of the predistorter can be somehow classified attending at some specific categories. An example of possible alternatives in the realization of the predistorter is depicted in Fig. 2.

Fig. 2. Classification of predistortion linearizers

According to the position of the predistorter in the complete transmitter structure, digital predistortion can be carried out at radiofrequency (RF), intermediate frequency (IF) or baseband (BB). Predistortion at IF and BB have the advantage of being independent on the final frequency band of operation, as well as they are more robust in front of adjustable front-end parameters. Besides, the cost and power consumption of ADC (analog to digital converters) and DAC (digital to analog converters) significantly decreases at low frequency operation. One drawback regarding the predistortion compensation at IF or BB is the increasing linearity requirements since the down-converters can introduce additional distortion. However, some downconverters (or I&Q modulators) may be evitable operating at IF and using software radio techniques, such as IF sampling. Another significant issue is

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Fig. 3. Block diagram of a transmitter with digital baseband predistortion

•

The digital data predistortion technique is custom tailored to specific digital modulation formats, so then, the predistortion function of data predistorters aims at compensating the data vector space (constellation). Therefore, the predistorter coefficients are optimized by means of minimizing the Error Vector Magnitude (EVM) that is, compensating the in-band distortion introduced by the PA. But digital data predistortion does not compensate the out-band distortion. In addition, data predistortion is not transparent to all different modulation formats, so it results only suitable for specific applications (and inappropriate from a more versatile point of view). On the other hand, digital signal predistortion is aimed at canceling both inband and out-band distortion, as long as the saturation level of the PA permits it. Moreover, digital signal predistortion is independent from the PA band of operation, class (A, AB, C...), power technology (SSPA, TWTA) and the most important, it is theoretically independent from the modulation format.

• •

an efficient model inversion procedure for the identification and adaptation of the baseband digital predistorter. an efficient implementation in the digital processing devices (DSPs or FPGAs) without an excessive computational cost.

PA models can be classified according to the type of data needed for their extraction in physical and empirical models [5]: o Physical models require the knowledge of the electronic elements that comprise the PA, their constitutive relations, and the rules describing their interactions. Physical models make use of a nonlinear model of the PA active device and other passive components to form a set of nonlinear equations relating terminal voltages and currents. The equivalent circuit description of the PA, whose topology is usually derived from the direct inspection of the real PA, permits accurate results very suitable for circuit-level simulation. However, their elevated simulation time and the need for a detailed description of the PA internal structure make them unsuitable for system level linearization purposes, such as digital baseband predistortion. o Empirical or behavioral models do not need an a priori knowledge of the PA internal composition, for that reason they are also known as black-box models. Their extraction relies on a set of inputoutput observations. Therefore their accuracy is highly sensitive to the adopted model structure and the parameter extraction procedure.

3 Power Amplifier Behavioral Models for Digital Baseband Adaptive Predistortion The versatility of the DPD linearizer has promoted a huge field of research for optimizing DSP architectures and algorithms. However, the cancellation performance of DPD can be degraded due to incorrectly modeled PA dynamics or because its sensitivity to memory effects generated in RF power amplifiers in broadband applications. Besides, computing time is a key factor to allocate the DPD algorithm in already existing DSP or FPGA devices keeping stuck other digital transmitter functions. Therefore, when designing digital baseband adaptive predistorters at least three major issues have to be taken into account:

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the need for accurate PA behavioral modeling capable to achieve the demanded requirements (in terms of BW, PAPR and memory effects reproduction).

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AM/AM and AM/PM characterization and can be expressed as: 2 αφ x(k ) (5) α A x(k ) f ( x ( k ) ) = f A ( x(k ) ) = φ 2 2 1 + βφ x ( k ) 1 + β A x(k )

PA behavioral (or black-box) models at system level are single input single output (SISO) systems. The extraction of PA behavioral model for DPD linearization purposes is carried out by means of input and output complex envelope signal observations. Fig. 3 shows a simplified transmitter block diagram presenting digital baseband predistortion linearization. We can observe the different type of signals (bandpass, baseband, analogue, digital) present after some of the transmitter subsystem blocks. The amplitude and phase modulated bandpass signal s ( t ) = A ( t ) cos (ω c t + θ ( t ) ) (1) can also be expressed in a polar form as, s (t ) = Re { x (t )e jω t } (2)

where α A , β A , αφ , βφ are constant parameters chosen to approximate the real PA characteristics. Regarding memoryless polynomial models, they are based on of pth order power series approximations: P

y ( k ) = ∑ γp ⋅ x ( k ) ⋅ x ( k )

p

(6)

p =0

where x(k) and y(k) are the PA input and output complex envelope at sample k, and γ p complex coefficients to characterize the PA behavior. In the Fourier series model, the output signal may be expressed from a complex Fourier series expansion of the instantaneous input signal. Applying complex Bessel series to the instantaneous complex Fourier series model, it is possible to derive a highly computable decomposed Bessel function based model suitable for both large and small signal behavioral modeling. And after some cumbersome developments [7], which include to fit the series components to the input signal band, the PA Bessel model may be simplified to

c

with x(t) being the complex envelope defined as,

x(t) = A(t)ejθ (t ) = A(t)cos(θ(t)) + jA(t)sin(θ(t)) = xI (t) + jxQ (t) (3) where xI (t ) and xQ (t ) are the in-phase and quadrature components of the complex envelope, respectively. Therefore, considering the complex envelope x(t) as a baseband signal described by a sequence of ideal pulses xI (t ) and xQ (t ) appearing at discrete times (k=1,2,3…) in a digital baseband environment, from (3) it is possible to express the complex envelope in the discrete form as,

∞

s0 (t ) = ∑ bk J1 (α kA(t ) ) e jωt

(3)

k =1

where A(t) is the signal amplitude, α is related with the dynamic range, and coefficients bk may be derived from measurements through a least squares extrapolation procedure.

x(t ) t = kT = xI ( kTs ) + jxQ ( kTs ) ≅ xI ( k ) + jxQ ( k ) (4) s Thus, in order to extract the PA behavioral models that will permit to design the digital predistorter at baseband, it is necessary to have input (x(k)) and output (y(k)) discrete complex envelope signals, as it is schematically shown in Fig. 3.

4 Memoryless Power Amplifier Behavioral Models Memoryless models, where the output signal is assumed to be a nonlinear function of the instantaneous input signal only, are suitable for narrowband applications, and usual approaches are the Saleh’s model, the polynomial model, Fourier series model or complex Bessel series [1][7]. Saleh proposed two general static polar functions to approximate the AM/AM and AM/PM envelope characteristics, initially applied to traveling wave tube (TWT) amplifier but widely used to also characterize solid state power amplifier (SSPA) ones. The polar representation corresponds directly to

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Fig. 4. Memoryless behavioral model

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Power Amplifier Behavioral Models with Memory Effects

The precise gain in RF power amplifiers presenting memory effects is not only a function of the input signal amplitude at the same instant, but also

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dependent on the recent history of the input-output signals as well.

may come from Hammerstein, Wiener or tree box models (Wiener-Hammerstein cascade) structures [11]. Hammerstein models are composed by a memoryless nonlinearity followed by a linear timeinvariant system: N

y Hamm ( k ) = ∑ α n f 0 ( x ( k − τ n ) )

(11)

n=0

where fo is a function describing the memoryless nonlinearity, which may be described with polynomials as: Fig. 5. Behavioral models with memory

y NMA ( k ) = ∑ f n ( x (k − τ n ) )

p

(9)

The augmented nonlinear moving average (NMA) PA behavioral model is an extension of the NMA model that introduces pairs of delayed samples of the input ( x ( k − τ i ), x ( k − τ j ) ) in order to improve nonlinear memory modeling

N

D

i =0

j =1

y NARMA ( k ) = ∑ f i ( x( k − τ i ) ) − ∑ g j ( y ( k − τ j ) )

(14)

with fi ( ⋅) and g j ( ⋅) being nonlinear memoryless

N

functions, and where τ i ( τ 0 = 0 ) and τ j ( τ ⊂ Ν ) are the most significant sparse delays of the input and the output respectively. Because of the IIR structure of

(10)

Besides, basic block diagrams for PA modeling

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(13)

Despite Wiener models cover a wider range of memory states than Hammerstein ones, its extraction results more complicated in comparison to the single step identification of Hammerstein models. A recent alternative are the Nonlinear Autoregressive Moving Average (NARMA) models [12], showing the advantage of the introduction of a nonlinear feedback path (infinite impulse response IIR) that permit relaxing the number of delayed samples to model the PA (Fig. 7). A NARMA NARMA model can be described as:

p =0

y AugNMA (k ) = ∑∑ f ij ( x( k − τ i ), x( k − τ j ) )

(12)

Fig. 6. Block diagram of a parallel-Hammerstein model

where the nonlinear function f n ( ⋅) is expressed by a polynomial function and where τ’s are the most suitable delays for describing the PA model.

i = 0 j =i

p =0

Both Hammerstein and Wiener models may be enlarged in order to achieve a more refined PA behavioral model identification. This is the case of the parallel-Hammerstein model depicted in Fig. 6.

(8)

n=0

N

n=0

p

 N  yWien ( k ) = f 0  ∑ α n x ( k − τ n )   n=0 

N

P

P

Wiener models (linear time-invariant system followed by a memoryless nonlinearity) are expressed as:

Coping with high speed envelope signals (presenting significant bandwidths) makes engineers reconsider the degradation suffered from memory effects. Time responses are convolved by the impulse response of the system and thus memory effects are no more irrelevant when predistortion type linearization is employed to cancel out the intermodulation sidebands. To develop amplifier models which include memory effects the amplifier has to be characterized using dynamic measurement systems. The most common dynamic nonlinear models considered in literature for identifying PAs and to compensate the PA dynamic nonlinearities by means of digital predistortion, are neural networks (NNs) [8] and simplified versions of the general Volterra series [9]. Memory polynomials or nonlinear moving average (NMA) models are one of the simplest models that take into account the nonlinear dynamic behavior of a PA and have been used for predistortion applications [10]. The input-output relation of a NMA model can be expressed as

f n ( x (k − τ n ) ) = ∑ α pn x ( k − τ n ) x (k − τ n )

N

y Hamm ( k ) = ∑ α n ∑ γ p x ( k − τ n ) x ( k − τ n )

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number of required coefficients in PA behavioral models may be reduced, finding a compromise between accuracy and computational complexity.

the NARMA model, it risks for instabilities which, if appears, they will avoid the use of the “linearized” PA since the consequent signal degradation and out of-band emissions (which become the PA behavior out of standardized emission mask). This challenge is solved by applying the small-gain theorem, which is an input-output stability method based in bounded norms [12]. As a result of the small-gain theorem application, the bounds of the NARMA model parameters which keep the system stable are obtained. This reduction of the complexity of the PA model acquires importance when the PA model is considered for real time linearization purposes, such in DPD linearization of PAs whose parameters may change in both large and small time scales (aging, temperature, load mismatching,...).

Acknowledgements: This work was supported by the Spanish Government (MEC) under Project TEC200507985-C03-02 and by the European Union Network Top Amplifier Research Group in a European Team (TARGET). References: [1] P.B. Kenington, “High-Linearity RF amplifier Design”, Aretech House, 2000 [2] K. Yamauchi et al., “A Novel Series Diode Linearizer for Mobile Radio Power Amplifiers,” IEEE MTT-S Digest, 1996, pp. 831-834. [3] Nielsen T.S., Lindfors S., “A 2.4 GHz MOSFET Predistorter for Dual-Mode Blue-tooth/IEEE 802.11b Transmitter”, Proc. of 20th Norchip Conference,November 2002, Copenhagen, [4] S. Andreoli, H. G. McClure, P. Banelli, and S. Cacopardi, “Digital linearizer for RF amplifiers”, IEEE Trans. Broad., vol.43, 1997. [5] J.C. Pedro, S.A. Maas, “A comparative overview of microwave and wireless power-amplifier behavioral modeling approaches,” IEEE Trans. MTT, vol. 53, pp.1150- 1163, 2005. [6] M. Isaksson, D. Wisell, D .Ronnow, “A comparative analysis of behavioral models for RF power amplifiers,” IEEE Trans MTT, vol. 54, 2006. [7] TARGET Report R2.2.E.2.6, “Power Amplifier Modelling Evaluation Report”. September 2006. (http://www.target-net.org). [8] T. Liu, S. Boumaiza, F. M. Ghannouchi, ”Dynamic behavioural Modeling of 3G Power Amplifiers using real-valued Time-Delay Neural Networks”, IEEE Trans. MTT., vol. 52, no. 3, pp. 1025-1033. 2004. [9] A. Zhu, T. J. Brazil, “An adaptive Volterra predistorter for the linearization of RF high power amplifiers”, MTT-IMS, vol. 1, pp. 461-464. 2002. [10] J. Kim, K. Konstantinou, “Digital predistortion of wideband signals based on power amplifier model with memory”, Electronics Letters, vol. 37, n. 23, pp: 1417-1418, Nov. 2001. [11] P. L. Gilabert, G. Montoro and E. Bertran, “On the Wiener and Hammerstein Models for Power Amplifier Predistortion,” APMC-05, vol. 2, pp.1191-1194, Suzhou, China, December 2005. [12] P. L. Gilabert, et alt, ”Multi-Lookup Table FPGA Implementation of an Adaptive Digital Predistorter for Linearizing RF Power Amplifiers with Memory Effects”, IEEE trans MTT, Vol. 56, n. 2, February 2008.

Fig. 7. Block diagram of the Nonlinear Auto Regressive Moving Average (NARMA) model.

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Conclusion

With the increasing bandwidth requirements in new telecommunication systems, behavioral models capable of reproducing PA memory effects as well as its nonlinear behavior become the most suitable for applying the envelope filtering technique in DPD linearization. However, these models add computational complexity to DPD algorithms which can derive in computational time enlargements and inefficient power consume of the DSP device. For that reason several simplified models derived from Volterra series (two and three box modeling) have been proposed as possible candidates for implementing the DPD function in a DSP. Moreover the NARMA model, which cannot be considered as a simplification of the Volterra series, introduces a conceptual feedback path (IIR filters) that can reduce the number of required coefficients to extract the model (or its inverse) and thus being even more computationally efficient. Related to this last issue, heuristic search algorithms can be used to find the optimal sparse delays contributing at describing PA memory effects. By using these search algorithms the

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