Another fundamental issue, especially in low-cost communications devices, is the nonlin- ear distortion. It is primarily produced by the different active components, in particular the amplifiers, and can heavily distort the signal. In principle, it stems from some form of clipping, which results in the highest signal peaks being compressed. In other words, when driven sufficiently close to saturation, the gain of the amplifier is smaller when the amplitude of the input signal is higher, and thereby the relationship between the input and output signals is in fact nonlinear. In the frequency domain, nonlinear distortion can be illustrated as shown in Fig. 2.7b, where it exhibits itself as spectral regrowth. Note that, in the context of IBFD transceivers, only the nonlinear distortion falling onto the signal band needs to be considered since the distortion falling out-of-band can easily be filtered out in the receiver. The out-of-band nonlinearities are an important consideration only in terms of the transmitter spectral emission mask, which is typically defined in the system specifications to limit the interference produced on to the adjacent channels [69]. Nevertheless, since the focus of this work is only on the inband distortion, the spectral emission requirements are not explicitly considered.
In a typical case, the main source of nonlinear distortion is the TX PA [P1, 18], [22, 144, 189]. The reason for this is the need for high power efficiency, while also having to
INBAND FULL-DUPLEX: BASIC PRINCIPLES AND ESSENTIAL SYSTEM MODELS
amplify the signal to the required transmit power level. These two requirements mean that the PA must operate close to its saturation point, which results in the nonlinear distortion of the waveform, especially with signals having high peak-to-average power ratio (PAPR) [105, 189]. Furthermore, even if the distortion is mild enough to fulfill the system specifications, it can still be extremely problematic for an IBFD device. For example, the strictest EVM requirement in the LTE specifications is 3.5% for the BS under the 256-QAM modulation scheme, which translates to a distortion component that is 29 dB weaker than the actual signal [68]. Making the reasonable assumption that the error component of a BS transmit signal is dominated by the PA nonlinearities [75, 76], this can be considered the highest allowed power level for the PA-induced nonlinear distortion under this modulation scheme. Hence, even with such a strict EVM requirement, the power level of the nonlinear distortion can easily be around 0 dBm at the TX output, meaning that in the RX chain it is likely orders of magnitude stronger than any signal of interest. What is more, when considering a lower order modulation scheme, the EVM requirements are less strict, meaning that the nonlinearities may be even stronger [68, 69]. This clearly indicates that nonlinear modeling of the TX PA is necessary in IBFD transceivers, as will be shown in more detail in Chapter 3.
In order to determine the power level of the nonlinear distortion more accurately, intercept points are commonly used. They describe the theoretical power level at which the nonlinear term of the considered order is equally powerful than the fundamental signal itself [79, p. 246]. This can then be used to calculate the power level of the nonlinear term for any reasonable component input power [P1]. As the 3rd-order nonlinearity is typically the dominant term, the 3rd-order input intercept point (IIP3) of the PA can be used to characterize the overall significance of the nonlinear distortion at the TX output. Then, the power level of the nonlinear distortion at the TX output can easily be approximated as follows [P1]:
pIMD,PA≈
p3 TX (iip3PA|kPA|2)
2, (2.7)
where pTXis the transmit power, iip3PA is the IIP3 of the PA, and |kPA|2is the gain of the PA, all in linear units.
There are also various methods for modeling the PA-induced nonlinear distortion on a waveform level, such as the Volterra series or the Wiener model [1, 173, 241]. However, to limit the complexity of the model, and to ensure efficient parameter estimation, the widely-deployed parallel Hammerstein (PH) model is adopted in this thesis. In principle, a PH model refers to a system with parallel static nonlinearities, each having their own filter that models the memory effects [213]. In this thesis, the static nonlinearities are chosen to be monomials, while the memory effects are modeled using finite impulse response (FIR) filters. Denoting the baseband-equivalent PA input signal by xin
PA(n),
its output signal can thereby be expressed with the adopted discrete-time PH model as follows [P5, 15, 18], [57, 101, 241]: xPA(n) = P X p=1 p odd MPH X m=0 hp(m) xinPA(n − m) p−1 xinPA(n − m), (2.8) 24
2.4 Analog Imperfections
where P is the nonlinearity order of the model, MPHis the memory length of the model, and hp(m) contains the coefficients of the pth-order nonlinearity. Note that it is sufficient
to include only the odd-order nonlinearities in the model when analyzing the inband distortion since the even-order terms fall outside the reception bandwidth [101]. This type of a model has been shown to be accurate for modeling a wide variety of practical PAs [15, 101, 173, 241], and in Chapter 5 it is shown to achieve high modeling accuracy also in the context of digital SI cancellation.
In addition to the nonlinear distortion produced in the transmitter, under some circumstances the RX chain can also distort the received signal in a nonlinear manner
[P1, 132], [10]. This typically occurs when the amount of SI suppression before the
receiver is too low, resulting in the saturation of the RX LNA. It is possible to model and attenuate also the RX-induced nonlinearities [132], [10], but the signal model becomes prohibitively complicated when also the PA is producing significant levels of distortion. Namely, then the overall coupling channel consists of the cascade of a nonlinearity, a wireless channel with memory, and another nonlinearity, resulting in an extremely large amount of nonlinear terms. For this reason, it is crucial to have sufficient RF cancellation performance since that will ensure that the power level of the receiver input signal is not high enough to produce significant nonlinear distortion in the LNA. This is also the underlying assumption in this thesis, and its validity is proven with the help of system calculations in Chapter 3 and further confirmed by the obtained measurement results reported in Chapter 5.