Chokri Jebali, Ghalid Idir Abib, Eric Bergault, and Ali Gharsallah

Digital wireless communications systems offer numerous advantages over their analog predecessors, including improved services and security.^1-4

But these digital systems also place greater demands on analog components in the system, including the power amplifiers (PAs) because of the complexity of the digitally modulated waveforms.⁵ To achieve the required levels of PA performance in terms of power, linearity, and efficiency, effective behavioral models are needed that include all the distortion-generating mechanisms within the amplifier as well as their memory effects. What follows is the first installment of a two-part article on a proposed behavioral model that accurately represents PAs processing signals in digital wireless communications systems.

In general, any amplifier driven into nonlinear operation will generate amplitude and phase distortion.⁶ These effects are characterized by amplitude-modulation-to-amplitudemodulation (AM-to-AM) conversion and amplitude-modulation-to-phasemodulation (AM-to-PM) conversion (AM-PM). These parameters represent variations in amplitude and phase transfer characteristics of a PA as it nears compression.⁷ Under these nonlinear conditions, an amplifier takes on a stepped-type output response, with each step corresponding to a higher order of distortion.⁸ The steps are known as spectral-regrowth sidebands or adjacent channel power (ACP).⁹ Intermodulation distortion (IMD), or spectral regrowth mainly impact adjacent channel interference.¹⁰ In fact, transmitter PA linearity is often specified in terms of demodulation error rates, rather than adjacent-channel spectral regrowth (ACSR).¹¹

The modulation format can often influence the design of a nonlinear PA. The p/4 differential quadrature phaseshift- keying (DQPSK) format is commonly used in North American Digital Cellular system (NADC) systems. It offers a compromise between the high channel capacity and low envelope amplitude variation, but differs from other modulation formats on its effect on amplifier distortion. Of course, literature sources note that a common envelope approach can be used to estimate distortion in PAs used for different modulation formats.^12,13 Mobile communications systems employ different types of modulation, including quadrature amplitude modulation (QAM) and 16-state QAM (16QAM). Linear modulation formats require linear amplification. The major reason for employing a linear scheme (i.e., narrow channel bandwidth) is to enhance spectrum efficiency. Some systems require the use of multiple channels simultaneously, using high-power devices such as traveling-wave-tube amplifiers (TWTAs) to support the channel bandwidths and power levels needed.¹⁴ Multicarrier transmitters use multiple narrowband carriers to transmit high-bit-rate data without an equalizer, typically with orthogonal- frequency-division-multiplex (OFDM) modulation.¹

Nonlinear behavioral models for PAs driving wideband-code-divisionmultiple- access (WCDMA) signals generally lack in accuracy compared to traditional linear device and amplifier models. Improvements have been made with recent nonlinear models based on Volterra series expressions.⁴ But amplifier memory effects are not included in these newer nonlinear models. To create a more accurate PA behavioral model for nonlinear situations, models must combine Vector Volterra models and memory polynomial models with sparse delay (MPMSD) to include memory effects as part of accurately predicting the power amplifier’s behavior.¹⁵ To improve upon the shortcomings of current nonlinear PA models, a test bed will be developed to study the effects of input versus output data on an amplifier model. The behavioral polynomial nonlinear PA model will be characterized when canceling the time delay to reach predistortion while taking into account memory effects for linearization.

The Volterra series is used as a general model for PA behavior. It requires a large number of basis functions. In the new model, special cases of Volterra series were applied.¹⁰ These were the envelope memory polynomial model, the conventional memory polynomial model, and the orthogonal memory polynomial model.¹¹ The model was based on PA measurements using QPSK test signals.

The mathematical formulation of the conventional polynomial model is given by:

and Eq. 2, where
x(n) = the input measurement,
y(n) = the output measurement,
k = the polynomial order,
M = the memory depth, and
a_ij = the polynomial coefficients.

When the complex gain of the device under test (DUT) is a function of the magnitude of the input signal, the envelope memory polynomial model can be described by Eq. 3.

The orthogonal memory polynomial model uses a set of basis functions to significantly improve identification accuracy by reducing the conditioning of the matrix to be inverted in the least-squares identification. The orthogonal model’s output is:

where U_ij is given by Eq. 5. The identification of the polynomial coefficients is given by the least-squares (LS) method in transforming the equations to matrix form. For the memory polynomial model (MPM), empirical (MPM) (EMPM), and orthogonal memoryless polynomial model (OMPM) approaches, it is possible to define a formulation as Eq. 6, where the basis function of the orthogonal model is given by Eqs. 7 and 8:

In the orthogonal polynomial model, the basis function φ is referred by ? in the whole of the expressions as

where φ^H is the Hermitian transpose.

In Eq. 14, memory effects introduced by thermal conditions, aging, and the presence of wideband signals:

where
k = the polynomial order,
Q = the memory depth order, and
a_ij = the polynomial coefficient.

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