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Overview of Nonlinear Effects in Wireless Communication Systems

This book is about the behavioral modeling of nonlinear communication circuits and their linearization based on a theoretical Volterra series approach. Nonlinear behavior is an inherent characteristic of electronic elements and devices, associated with the function they perform in a radio frequency (RF) communication system. To be successfully captured by a remote observer, the information-carrying signal must be strongly amplified at the cost of giving rise to nonlinear distortions. In the same way, the generation of carriers or the process of incorporating information into the carrier signal is realizable, thanks to the nonlinear operation of the different modules in the transmitter. The price for these valuable features is the generation of nonlinear imperfections in the signal sent, with the appearance of two adverse collateral consequences: provoking misinformation in the recipient of the signal and interfering with other users of the system. The central objectives of this book are the study of these nonlinear problems in wireless communication systems and the research of techniques to compensate for nonlinear impairments in order to ensure efficient, error-free transmission without interference to other users.

1.1 Wireless Communication Systems

1.1.1 Transmitters and Receivers

The scenario given by a typical wireless communication transmitter-receiver link is shown in Figure 1.1. The binary data with the information to be transmitted is converted to a baseband analog signal used to modulate an RF carrier and then amplified and radiated using the transmitter antenna. On the receiver side, the signal is captured by the receiver antenna and demodulated before its conversion to the discrete-time received sequence.

Figure 1.1 Block diagram of a typical wireless communication transmitter-receiver link.

The theoretical assumption of linear operation in wireless networks is only an approximation because the transmitters are built with several blocks, namely, modulators, mixers, power amplifiers, etc., whose electronic circuits are essentially nonlinear. In practice, undesired nonlinear effects produced in transmitters, mainly in the power amplifier, degrade the system performance and cause difficulties in meeting the stringent requirements set by the standardization entities, such as spectral masks and dispersion in the constellation.

Modern wireless communication systems are designed to operate with digitally modulated signals that have large bandwidths and high peak-to-average power ratios. Since the nonlinear behavior of the system is heavily dependent on the input signals employed, advanced knowledge of the digital world is more than advisable for the modern RF engineer.

Over the past few decades, wireless communication systems have been increasing data rates and capacity as a consequence of more sophisticated and efficient cellular networks. New generation systems employ highly spectrally efficient modulation schemes and solutions, such as orthogonal frequency division multiplexing (OFDM) and multiple-input multiple-output (MIMO). The fifth generation (5G) systems make use of both solutions, together with other technical developments that they have introduced in radio access networks. Furthermore, the trend in 5G systems to increase the use of frequency bands over 6 GHz, also forecasted for beyond 5G and 6G systems to satisfy the ever-increasing demand for connectivity, will also lead to signals more sensitive to nonlinear effects.

Noticeable nonlinear effects are generated in the circuits by the envelope variations of these signals. The nonlinear operation cannot be easily described in an analytical way; therefore, optimized designs are complex. In addition to the changes in amplitude and phase shifts typically observed in linear systems, spurious components are generated in nonlinear circuits, distorting the amplifier or mixer behavior. Among the effects of nonlinear distortions, intermodulation distortion and spectral regrowth should be taken into account since they cannot be eliminated by filtering and produce detrimental adjacent channel interference (Maas, 2003).

It is also worth noticing that, today, circuits operating in the RF range coexist with low-frequency baseband signal processing. This baseband signal processing has undergone a substantial evolution over the last few years that has led to complex modern systems. The study and prediction of the behavior of RF systems can benefit from the application of signal-processing techniques.

The present evolution toward more sustainable communications involves the search for energy-efficient transceivers, with the power amplifier being the most critical subsystem of the transmitter in terms of power consumption. In this context, the performance enhancement of a wireless communication system when its efficiency and power consumption are optimized is clear considering that these ubiquitous networks are constituted by a multitude of base stations, each one with a transmitter and a nonlinear power amplifier. However, we should recall that the power amplifier is a source of undesired nonlinear effects, more notable especially as its efficiency increases. Therefore, as RF engineers, we want to study and understand these nonlinear effects. We also want to compensate for nonlinearities and construct a linearized transmitter (predistortion). In other cases, we want to equalize the signal in the receiver (postdistortion) in the presence of a high-level noise.

1.1.2 Real-valued Continuous-time RF Signals and Complex-valued Discrete-time Baseband Signals

RF transmitters of communication systems based on modern wireless standards, like 4G, 5G, and beyond, generate real-valued continuous-time bandpass signals with a frequency response that occupies a limited bandwidth centered around the carrier frequency . The trend is to use a broad signal bandwidth, but in all cases, the condition is satisfied1. The transmitted bandpass signals are the result of modulation, i.e., the incorporation of the baseband signal information to an RF carrier.

A common representation of a bandpass signal is2

where is the signal's complex envelope, an equivalent low-pass signal, and is the angular frequency of the carrier. The output of a bandpass linear system centered at , with a signal applied at the input, is given by the convolution:

where is the real-valued impulse response of the system. The bandpass radio communication channel is an example of a linear system described by (1.2), whereas this linear convolution is insufficient to explain the nonlinear behavior of the power amplifier, for example. The extension of the linear convolution (1.2) to the case of nonlinear systems is the Volterra series (Volterra, 1959), a major topic of discussion throughout the following chapters of this book.

Similar to the equivalent low-pass signal, the bandpass radio communication channel can be modeled with its equivalent low-pass channel impulse response to obtain the output complex envelope as the complex-valued convolution , formulated analogously to equation (1.2). It should not go unnoticed that this complex-valued convolution is essentially different to the real-valued convolution (1.2), because it involves the four convolutions of the real and imaginary parts of the equivalent low-pass signal and the impulse response .

All the interesting information is contained in the complex envelope, and as a consequence, the RF signal bandwidth can be controlled in the baseband blocks of the transmitter. This is why the low-pass equivalent representation offers not only a more convenient perspective to the analysis of a communication system, but also the possibility of counteracting in the baseband blocks for the alterations induced in the transmitted signal by the radio channel impairments. Practical techniques, e.g., OFDM, coding, equalization, and MIMO, are frequently implemented at baseband to maximize spectral efficiency in a reliable communication.

Formally, compensation of linear impairments in the radio channel can be managed with an RF linear block either in the transmitter (pre-compensation) or in the receiver (post-compensation). However, the benefits of low-frequency modules compared to high-frequency circuits suggest that a baseband filter involving a complex-valued linear convolution would be the best alternative. In this case, the pre-equalizer is the inverse module of the radio channel that introduces the necessary distortion to compensate for the transmission's linear impairments. Therefore, the pre-equalizer output can be modeled with a linear convolution enunciated likewise in the baseband complex domain and in the RF real domain. The usefulness of a predistorter module to compensate for nonlinear impairments in a communication system can be substantiated similarly, giving also a justification for the baseband solution and bearing in mind an important reflection. Recalling that although convolutions for real-valued and complex-valued linear systems are strictly different because the complex-valued convolution involves four real-valued convolutions, both linear convolutions have the same form. This is not true for the case of nonlinear systems, and therefore, the reasoning behind the equivalence in the form of baseband and bandpass RF representations cannot be extended to nonlinear models. This is a...