On the reasonableness of nonlinear models for high power amplifiers and their applications in communication system simulations

High power amplifier (HPA) models with inherent nonlinearities play an important role in analysis and evaluation of communication system performance in both theoretical and practical aspects. However, there are not so much discussions on the suitability to the use of such models in simulating HPA nonlinearity in communication systems.

ON THE REASONABLENESS OF NONLINEAR MODELS FOR HIGH POWER AMPLIFIERS AND THEIR APPLICATIONS IN

COMMUNICATION SYSTEM SIMULATIONS

Nguyen Thanh1,*, Nguyen Tat Nam2, Nguyen Quoc Binh1,3

Abstract: High power amplifier (HPA) models with inherent nonlinearities play

an important role in analysis and evaluation of communication system performance

in both theoretical and practical aspects However, there are not so much discussions on the suitability to the use of such models in simulating HPA nonlinearity in communication systems In this work, we investigate the reasonableness of well-known nonlinear models and propose two models that are both analytic and better than Cann’s new model in terms of approximating to the real-world data Examples with specific testing signals verify the relevance of the arguments and point out suitable alternatives for use

Keywords: High power amplifier; Nonlinear modeling; Nonlinear distortion simulation.

1 INTRODUCTION

Generally, for many communication systems such as satellite or mobile communications, power and/or bandwidth efficiencies are among the leading interests On the other hand, for high power efficiency, amplifiers behave nonlinearities unignored Nonlinear characteristics show an important influence for small-signal stages of a receiver since intermodulation products can strongly interfere with the desired signals However, with less dealing to the power efficiency problem than performance considerations, the small-signal amplifiers then should be well linearized Therefore, studies on the nonlinear characteristics commonly focus on the high power amplifiers (HPAs)

Generally, there is a tradeoff between HPA’s maximum power efficiency that requires pushing its operating point well into saturation, and minimizing nonlinear distortion, namely, demanding that the HPA operates well below saturation for diminishing spectrum regrowth, nonlinear interference (ISI) and interchannel interference (ICI) [18]-[20] This problem has been discussed widely manifesting as the tradeoffs between output power back-off (OBO), linearization, adjacent channel power ratio (ACPR), However, these works mostly based on the envelope models while rarely considered the instantaneous models

Different techniques are employed to operate the HPA at its highest possible power efficiency but satisfying the linear specifications, at the same time If the designed HPA does not fulfill the ACPR specification for a desired operating frequency, linearization techniques are usually applied to improve its linearity These procedures require extensive simulation work and reliable large-signal model is indispensable Similarly, other complex efficiency enhancing HPA design techniques also need large-signal model.As a simple method, the HPA nonlinear characteristics are usually measured at separated points based

on one or two unmodulated carrier(s) Then, for system analysis or simulation purposes, interpolation/extrapolation should be carried out to retrieve the desired characteristics For these reasons, the approximated close-form model is a very convenient tool for the replacement However, for a long time, the suitablity of using such nonlinear models in simulating communication systems with HPA nonlinearity is not much investigated This could at least create a significant gap between theoretical research results and realities or more severely, might produce invalid research results

Looking back in the past, in 1980, Cann [5] proposed an instantaneous nonlinearity model for HPA with the convenient feature of variable knee sharpness, mostly suitable for both theoretical analysis and simulation However, until 1996, Litva [3] shown that this model give incorrect results for intermodulation products (IMPs) in the two-tone test Four years later, Loyka [9] diagnosed the reason: non-analyticity Other publications showed that no problem exists with typical real-world signals

Recently, Cann [6] improved the original instantaneous model, totally eliminating the problem with minimal complexity augmentation However, to investigate its applicability as

an envelope model for simulating nonlinearities in communication systems, there need careful analyses since the usage of the instantaneous models is quite different to that of the envelope models Moreover, with a rather structural form of formulation, the Cann’s new model is inherently less accurate in approximating to the real-world data Thus, there is a gap

to fill in by more suitalble models that are analytic and better approximate to the reality Therefore, this work investigates the reasonableness of the current widespread-used nonlinear models, and proposes two models that are both analytic and better than Cann’s new model in terms of approximating to the real-world data Examples with specific testing signals verify the relevance of the arguments and point out suitable alternatives for use The rest of this paper is organized as follows The Cann’s original instantaneous model and improved version are introduced and analyzed in Section 2, emphasizing on the defects of non-analyticity and asymmetry Focusing on the same targets, Section 3 carries out analyses for two proposed models and other extensively used envelope models Examples with numerical results are shown and discussed in Section 4 revealing suitable models for use Section 5 concludes the achievements

2 CANN'S MODEL FOR INSTANTANEOUS SIGNALS

Cann's original model

To represent a signal passing through an HPA, in 1980, Cann [5] proposed the instantaneous nonlinear model with variable knee sharpness

os

os

os

·sgn( )

| |

| |

y



(1)

where, y is the output voltage, x is the input voltage, g is the small-signal (linear) gain,

os

A is the output saturation level and s is the sharpness (smoothness) parameter This is one of the oldest nonlinear models for representing HPA [1]

However, until 1996, Litva [3] found that this model gave incorrect results for the third and higher order IMPs in the two-tone test Four years later, Loyka [9] discovered that the reason was the use of modulus (|.|) function in (1), some of whose derivatives at zero do not exist, are undefined, or are infinite In other words, the function is not analytic, despite the deceptively smooth appearance of the plotted curves

Incidentally, in 1991, Rapp [8] introduced a complex envelope model for solid-state power amplifiers (SSPAs) that resembles to the Cann’s instantaneous model except for the modulus operator in the denominator and the exponent of 2s instead of s However, at the best of our knowledge, there are not so much discussions on the defect of this commonly used model Detail analysis on this topic will be given in the next section

Cann's new model

Based on the magnitude Bode plot of a simple lead network transfer function

(1 jω) / (a jω) which is analytic and symmetric regarding to variable ω, Cann [6] suggested the new nonlinear instantaneous model in the scaled normalized form as

os

os

( / 1) os

os ( / -1)

1

-1

s gx A

s gx A







 (2)

with variables y, x, and parameters g, Aos, and s have the same meanings as what are

in the original model (1)

It is not difficult to show that the derivatives of new model’s (2) exist and well behave, even with fractional s The reasonableness of the third and fifth order IMPs for the two-tone test simulation using this model is illustrated in figure 1 Here, the sharpnesses s

vary in a quite large range revealing the model’s effectiveness

It is observed that these lines have the expected slopes as what happening in a real-world experiment: 3 dB/dB for third order (in figure 1.a)) and 5 dB/dB for fifth order (in figure 1.b)) Moreover, the IMPs’ slopes do not change for all sharpnesses This confirms the suitability of the new Cann’s model (2), yielding simulation results conforming to what happening in reality Therefore, model (2) totally eliminates the shortcomings of the previous one This is the analyticity and symmetry of the original lead network transfer function to resolve the problem

-80 -70 -60 -50 -40 -30 -20 -10 0

-280

-240

-200

-160

-120

-80

-40

0

Input [dB]

s = 3

s = 5

s = 9

-350 -300 -250 -200 -150 -100 -50

Input [dB]

s = 3

s = 5

s = 9

(a) (b)

Figure 1 IMPs by the new Cann’s model (2): a) Third order; b) Fifth order

3 ENVELOPE MODELS

Envelope representation of bandpass signals

Practically, to comply with spectral regulations, a communication system with nonlinear HPA often has a bandpass zonal filter that restricts the output to the first spectral zone, suppressing all harmonics and even-order IMPs Such a system is referred as narrowband or bandpass, meaning that the bandwidth is considerably less than the center frequency This attribute allows huge saving of computation since the required sampling rate is then determined not by the highest frequency of the signal but by its bandwidth (of course plus a suitable redundance for significant IMPs) The resulting model is the

lowpass equivalent representation of the bandpass system and is regarded as envelope model

A narrowband radio-frequency (RF) signal can be represented as

[ ( )]

( ) ( ) cos[ ( )] Re[ ( ) j t t ]

v t A t  t  t A t e  

   , (3) where, A t( ) is the amplitude modulation (AM) component, and ( )t is the phase modulation (PM) component, both varying slowly regarding to the carrier frequency  When being observed in a reference plane rotating at the carrier frequency, the resulting

signal is complex envelope

( ) ( ) ( ) j t ( ) cos ( ) ( ) sin ( )

x t  A t e  A t  t  jA t  t (4)

It is noteworthy that the carrier  disappears but all modulating information (carried in both amplitude and phase) still exists in (4)

Envelope model characteristics

The envelope model is characterized by its complex transfer function F A( ) y x/  ( )

( ) jF p A

a

F A e , including the AM-AM transfer function F Aa( ), output amplitude as a function of input amplitude, and the AM-PM transfer function F A p( ), phase shift as a function of input amplitude, all for a single frequency signal At rather low frequencies and small bandwidth, SSPA previously was considered as having little or no AM-PM and

a constant transfer function over the passband [11] However, at higher frequencies and larger bandwidth, this assumption is no longer valid [13]-[17]

Actually, measurements of these transfer functions are usually made at only a discrete set of points; therefore, to simulate the nonlinearity at a specific operating point, generally, the input-output relation is usually interpolated from measured data This can be carried out with great accuracy using series expansion or splines,… but a closed-form model can provide a convenient approximation and is often accurate enough

Saleh model

0

0.2

0.4

0.6

0.8

1

1.2

Normalized input magnitude

Saleh (5) Mod Saleh (11) Mod Ghorbani (13) Rapp (7)

-20 -10 0 10 20 30 40 50 60

Normalized input magnitude

Saleh (6) Mod Saleh (12) Mod Ghorbani (14) Mod Rapp (15)

(a) (b)

Figure 2 Characteristics of typical nonlinear models: a) Amplitude; b) Phase

In 1981, Saleh, a researcher working at Bell Labs in Crawford Hill, introduced a close-form model for traveling wave tube amplifiers (TWTAs) [7], which then has been widely used since it includes both AM-PM and AM-AM with typical turndown after saturation These AM-AM, AM-PM are formulated as:

( ) 2

1

a a

a

A

F A

A







 , (5)

2 2 ( )

1

p p

p

A

F A

A







 , (6)

where, A is the input amplitude, F Aa( ) is the output voltage, F A p( ) is the phase shift,

a

 is the small-signal (linear) gain, together with   a, p, p forming the shape of amplitude and phase conversion curves, a (a / 2Aos)2, Aos is the output saturation level This model is illustrated in figure 2 with normalized linear gain and input saturation level, a  1, Aos  1 [V] This figure also illustrates other typical AM-AM and AM-PM characteristics which are then discussed below

Saleh reminded that the amplitude A might be negative, thus, (5) must be an odd function Noting that the Saleh model does not support adjusting the knee sharpness of AM-AM characteristic Otherwise, the curvature of (5) is too smooth regarding to the typical SSPAs’ AM-AM characteristics, which also do not fall down after saturation

Rapp model

In 1991, in a work studying the effects of nonlinear HPA in digital broadcasting system, Rapp proposed an envelope model with variable knee sharpness for SSPAs as [8]

1/ 2 2

os

( ) 1

gA

F A

gA A



, (7)

where, A is the input magnitude, F Aa( ) is the output mangitude, g is the small-signal (linear) gain, and s is the curve’s sharpness

0 0.2 0.4 0.6 0.8 1 1.2

Input voltage [V]

Ideal limiter Rapp, s = 1 Rapp, s = 1.4 Rapp, s = 3 Rapp, s = 

Figure 3 Amplitude characteristics of the Rapp model with different sharpnesses

It is noteworthy that this model assumed zero AM-PM conversion and by changing the sharpness parameter s, the AM-AM characteristic could have any curvature Further, (7)

is only odd (Saleh’s condition) for integer s

Several examples of (7) with different knee sharpnesses s are illustrated in figure 3 with normalized linear gain and output saturation level, g  1, Aos  1 [V] In addition to this, the normalized characteristic curve of the ideal limiter is included for reference

purpose This is an upper bound for any real-world amplifiers (with approximated exception of ideal predistorter-amplifier combination [17], [18])

Incidentally, the Rapp’s model resembles to the instantaneous model (1) excepting the absence of modulus operator in the denominator Thus, it seems to avoid the problem of (1) for the suitability of IMPs resulted by simulation, but this is not the case The Rapp’s model has been widely used for roughly a quarter of century without any notation for its reasonableness and also its suspicious results until the publication of Cann [6]

Thorough investigation leads to the conclusion that the problem of (1) only manifests with signals that have their magnitude distribution concentrating around zero, such as the

signal used in the two-tone test For real-world signals like FSK, PSK, QAM,

M-APSK, OFDM,… the Rapp’s model behaves almost perfectly well

Therefore, resembling to the case of instantaneous models, all envelope AM-AM models should ideally be odd and analytic over the expected amplitude range An envelope model, which is asymmetric and is not analytic at zero, should be used with caution and only for signal waveforms that are sufficiently complex to have a wide amplitude distribution However, non-analytic model is not a serious defect, because typical real-world signals with high spectral efficiency have large amplitude distribution It is well known that signal should be noise-like for maximizing the channel capacity

Cann’s new model

0 5 10 15 20 25 30 35

Input [V]

Data Cann (2) Rapp (7) Polynomial (8) Polynomial (9) Polysine (10)

1.15 1.2 1.25 1.3 28.5

29 29.5

20 21 22

Figure 4 Rapp, Cann, polynomial and polysine models’ amplitude characteristics

fitted to measured data

Although originally developed as an instantaneous model, (2) can be used equally as an envelope model This should find broad applications, like Rapp model, it has adjustable knee sharpness and does not turn down after saturation But, unlike the Rapp model, it is analytic everywhere and therefore valid for any signal waveform Moreover, if the phase convesion is significant, an AM-PM function, such as Saleh’s (6), can be included Resembling to the Rapp model (7), envelope model (2) could support any curvature, especially in the region above s 2.5, suitable for AM-AM characteristics of most SSPAs [17] The approximations of model (7) and model (2) to the real-world data are verified by curve fitting of these functions to the measured data from the L band Quasonix 10W amplifier [12] Results are, for Rapp model (7): g 29.4, A os 30[V], s 4.15, for the new Cann model (2):g 29.4, A os 30[V], s 8.9, [6] For this particular

HPA, Rapp model is little better fitted than Cann model Figure 4 illustrates these fittings with the inclusion of other approximated curves discussed next

Polynomial models

Considering the measured data in figure 4, it is not difficult to recognized that there is a simple yet efficient method approaching the close-form characteristic function by approximation using polynomials In this case, the complex envelope nonlinearity

F A  y x can be represented by a complex polynomial power series of a finite order

N such that

1

   , (8) where, k P[ ] | |x  x k1 x are the basis functions of the polynomial model, and ak are the model’s complex coefficients

Table 1 Coefficients of polynomial models (8) and (9)

Model a1 a2 a3 a4 a5 a6 a7 a8 a9

(10) 30.02 -8.665 33.68 -40.19 12.39 0 0 0 0

(11) 28.60 0 8.310 0 -15.06 0 6.257 0 -0.872

Obviously, model (8) is not analytic at A| | 0x  by the existence of modulus operators However, if even order coefficients a2k vanish, then, for real-valued signals

( )

x t , (8) turns into the odd order polynomial model of the form

| |

  (9) Model (9) is clearly analytic at A| | 0x  and is used as a counter example to model (8) in the applications section below The measured data of the L band Quasonix 10W amplifier is then used to fit the polynomial models (8) and (9) with the same number of coefficients N 5 Figure 4 depicts the approximated characteristics with parameters shown in table 1

It is not difficult to show that at large enough order, polynomial models are better fitted

to the real-world data than Rapp model (7) and Cann model (2) Further, with the same N,

higher order polynomial in (9) is smoother than lower order one in (8) resulting better fitting performance for the sooner

Polysine model

It can be seen that the sine/cosine functions are distinctly better than polynomial ones in terms of both analyticity and smoothness Thus, while remaining to be analytic, the sooners are better fitted to the real-world data than the laters Based on this argument, we propose the nonlinear model of the form

1

N

k k k



 , (10) where, ak and bk are correspondingly the amplitude annd phase coefficients The introduction of bk lets the function better addapting to the fitting data, thus improving the approximation performance

Using the Matlab curve fitting tool, (10) is fixed to the AM-AM characteristic of the L band Quasonix 10W amplifier data [12] in figure 4 resulting in the parameters listed in table 2

Table 2 Coefficients of polysine model (10)

a k 30.73 -0.6586 -0.1061 0.00955

4 0.1859

b k 1.045 5.312 12.91 18.61 8.107 The fitting performances of these five models are quantified using Square Error Sum (SES) measure and are compared in table 3 Odd-order polynomial model (9) and polysine model (10) are both analytic and much better fitted to the real data than Cann model (2) This is illustrated in figure 4 with sub-figures focusing on segments with significant differences where the data is rather harder to fit The better fitting performance is the closer to the data these curves approach With almost one order of magnitude better in SES than the rest, the polysine model’s curve always coincide to all data points The fitting performance of these models will reflect in the nonlinearity simulation results that are then discussed bellow

Table 3 Fitting performance (SES e2) of five models

Model Cann

(2)

Rapp (7)

Polynomi

al (8)

Polynomi

al (9)

Polysine (10) SES 1.786 0.963 0.533 0.346 0.032

Other models

Beside the AM-AM characteristic, updated envelope models for SSPAs at higher frequencies and larger bandwidth all consider the AM-PM conversion and generally better fit to the measured data than previous models However, it is not difficult to see that models discussed below are not analytic or symmetric at A 0 for most of the parameter sets and thus problem of (7) still exists The characteristics of these models are graphically illustrated in figure 2 for comparison purpose

Modified Saleh model

The modified Saleh model [13] was proposed for popular LDMOS (Laterally diffused metal oxide semiconductor) power amplifiers (PAs), that are very common for the base station (BS) amplifiers of 2G, 3G and 4G mobile networks (in the L, S, C bands) The AM-AM and AM-PM conversion functions are

3

( )

1

a a

a

A

F A

A









, (11)

4 3

( )

1

p

F A

A







, (12) where, a  1.0536, a  0.086, p 0.161, p 0.124 is a typical parameter set

Modified Ghorbani model

The modified Ghorbani model [14] that is suited for GaAs pHEMT FETs (Gallium arsenide pseudomorphic High-electron-mobility transistor Field-effect transistor) PAs that are operating at frequencies upto 26 GHz (K band) and are dominant in terms of production technologies and market shares compared to other power semiconductor techlogogies This model proposed the following charactertistics

2

1

4

( )

1

x a

F A x A x A

x A







 , (13)

2

1

4

( )

1

y p

F A y A y A

y A







 , (14) where, the model parameters are given by x 1 7.851, x 2 1.5388, x  3 0.4511,

4 6.3531

x  , y 1 4.6388, y 2 2.0949, y  3 0.0325, y 4 10.8217

Modified Rapp model

The modified Rapp model [16] was introduced for GaAs pHEMT/CMOS (Complementary metal-oxide-semiconductor) PA model at 60 GHz band, the new band for communication industry, with AM-AM function of (7) and AM-PM described as

1

2 ( )

1

q

A

F A

A









, (15)

where, parameter set are g 16, A os 1.9, s 1.1,   345,  0.17, q1 q2  4

4 APPLICATIONS

This section describes the applications of envelope models investigated above for representing nonlinear HPA in communication systems and analyses typical experiments with test signals having discrete and continuous spectra to reveal their applicability and reasonableness

Representation of envelope model

Consider the finding of IMPs in a two-tone test with a signal consisting of two equal-amplitude unmodulated sinusoid waveforms at frequencies f1 and f2  f1 These testing signal could be equivalently regarded as a double-sideband suppressed carrier AM of the form

1

x t  A  f t   f t A  f t f t , (16)

where, fm 12( f2 f1) is the modulating frequency, fc  12( f2  f1) is the (center) carrier frequency Waveform (15) with f 1 7 [Hz], f 2 10 [Hz] is illustrated in figure

5 It is observed that the carrier fc manifests inside the envelope and is the average of f1

and f2, while the envelope is the modulating signal at frequency fm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2

-1 0 1 2

Figure 5 Two-tone signal waveform with f 1 = 7 [Hz], f 2 = 10 [Hz]

With the 90o phase shifting, xinst( ) t in (16) could be recast as

inst( ) = 0sin(2 m ) si n( 2 c )

x t A  f t  f t (17) Therefore, its envelope form is

env( ) 0sin(2 m )

x t  A  f t (18)

( )

A t

( )

( ) j t

( )t



( )

V t

( )t

 ( )

( )

( ) j t

Figure 6 Polar envelope model block diagram

Because the envelope model requires non-negative input, thus, the sinusoid waveform

of (18) is decomposited to the polar form as

env( ) ( ) j t 0| sin(2 m ) | j t

x t  A t e  A  f t e , (19) where,

0 ( ) | sin(2 m ) |

A t  A  f t , (20)

( ) 0, sin(2 ) 0

, sin(2 ) 0.

m t

m

f t e

f t



 





r

(21)

In other words, the amplitude component A t( ) is the full-wave-rectified sinusoid, and the phase component ( )t is the 180o square wave

When passing through the envelope model, the amplitude component is input to the model, while the phase component is bypassed as depicted in figure 6 [1] The distorted amplitude output is then combined with the phase part, resulting the output waveform for analysis If AM-PM conversion is included, then the distorted phase is added up to the input phase ( )t before combining