In this section, we discuss the main sources of non-idealities in the self-interference signal and we provide measurements from our testbed that clearly demonstrate the existence of most of the impairments that have been previously considered in the bibliography. More specifically, we confirm the presence and the effect of phase noise [17-20], IQ imbalance [21,22] and RF non-linearities [22,24,25]. More importantly, however, we observe the existence of baseband non-linearities with significant power. We show that these non-linearities are highly consistent with non-linearities stemming for the transmitter DACs, and we provide a corresponding DAC non-linearity model that can be used for digital self-interference cancellation.

We denote the discrete time digital baseband signal by *x*[ *n*] and the continuous time analog baseband signal by \({\tilde {x}}(t)\). The upconverted analog signal is denoted by *x*(*t*), and the amplified upconverted analog signal is denoted by \({\hat {x}}(t)\). The downconverted analog self-interference signal at the receiver is denoted by \({\tilde {r}}(t)\), and the digital baseband self-interference signal is denoted by *r*[ *n*]. For simplicity, we omit the time indices *n* and *t*, unless strictly necessary (e.g., to denote a delay as in the case of phase noise), as they will always be clear from the context. Figure 3 shows a block diagram of a full-duplex transceiver node, with all important analog front-end components and signal terms identified.

### 4.1 Phase noise

The upconversion of the baseband signal to the carrier frequency *f*
_{
c
} is performed at the transmitter by mixing the baseband signal with a carrier signal. The oscillators that are used to generate the carrier signal suffer from various impairments, the most significant of which is phase noise. Thus, instead of generating a pure tone at frequency *f*
_{
c
}, i.e., \(\phantom {\dot {i}\!}e^{j2\pi f_{c} t}\), the generated tone is actually \(\phantom {\dot {i}\!}e^{j(2\pi f_{c} t+\phi (t))}\), where *ϕ*(*t*) is the random phase noise process. The downcoversion process at the receiver is also affected by phase noise, since it uses a similarly generated carrier signal.

The effect of phase noise on full-duplex transceivers has been extensively studied [18-20]. In order to summarise and illustrate the effect of the phase noise, we assume for the moment that all parts of the transceiver are ideal, except for the oscillators that generate the carrier signal. Moreover, for illustration purposes, the self-interference channel is assumed to introduce a simple delay, i.e., it can be represented as *δ*(*t*−*Δ*
*t*). Such a delay can for example arise from acoustic wave bandpass filters in the receive chain. At the transmitter, phase noise is introduced during the upconversion process, so the transmitted RF signal is:

$$\begin{array}{*{20}l} {\hat{x}} & = \Re\left\{{\tilde{x}} e^{j(2\pi f_{c}t+\phi_{\text{Tx}}(t))}\right\}, \end{array} $$

((1))

where *ϕ*
_{Tx}(*t*) denotes the phase noise process of the oscillator used by the transmitter. At the receiver, phase noise is introduced during the downconversion process:

$$\begin{array}{*{20}l} {\hat{r}} & = \text{LPF}\left\{{\hat{x}} e^{-j(2\pi f_{c}t+\phi_{\text{Rx}}(t-\Delta t))}\right\} \end{array} $$

((2))

$$\begin{array}{*{20}l} & = {\tilde{x}} e^{j(\phi_{\text{Tx}}(t)-\phi_{\text{Rx}}(t-\Delta t))-2\pi f_{c} \Delta t}, \end{array} $$

((3))

where LPF denotes a low-pass filter that removes the copies of the signal around 2*f*
_{
c
} and −2*f*
_{
c
} and *ϕ*
_{Rx}(*t*) denotes the phase noise process of the oscillator used by the receiver. If the transmitter and the receiver use independent oscillators, the *ϕ*
_{Tx}(*t*) and *ϕ*
_{Rx}(*t*) processes will be uncorrelated. However, in full-duplex transceivers, the transmitter and the receiver are typically co-located and can physically share the same oscillator. Thus, *ϕ*
_{Tx}(*t*)=*ϕ*
_{Rx}(*t*), and we denote the common phase noise process by *ϕ*(*t*). Due to the delay introduced by the transmission channel, the phase noise instances experienced by the signal at the transmitter and the receiver mixer are not identical. However, it is evident that if the delay is such that *ϕ*(*t*) and *ϕ*(*t*−*Δ*
*t*) are highly correlated, then sharing the oscillator can significantly reduce the effect of phase noise in the received signal after the mixer [18].

In order to demonstrate the improvement obtained by sharing the oscillator between the transmitter and the receiver mixers, we perform a one-tone test on our testbed and we examine the received self-interference signal. We use the lowest possible transmit power setting (i.e., −10 dBm) in order to minimise non-linearities arising from the amplifier of the transmitter. An indicative spectrum of the received self-interference signal is presented in Figure 4a. We observe that the received signal has significant spectral content around the transmitted tone, indicated by (i), with the most powerful components lying approximately 46 dB below the power of the tone. We note that the two independent oscillators use the same 10 MHz reference signal, but this is not sufficient to reduce the effect of phase noise, which is caused by the phase-locked loop (PLL) that generates the actual carrier signal from the reference signal. We also observe numerous tones arising from other non-linearities, which will be explained in the following sections. In Figure 4b, we present an indicative spectrum of the received self-interference signal when the transmit and receiver mixers use the same oscillator. In this case, the strongest component of the spectral content resulting from phase noise lies approximately 70 dB below the received tone. We observe that the phase noise induced noise floor will lie significantly below the noise floor introduced by the remaining non-linearities.

### 4.2 Baseband non-linearities

In Figure 4a, we observe that numerous tones have appeared in the received signal apart from the transmitted tone. The eminent tones on the positive side of the spectrum appear at integer multiples of the transmitted tone frequency. Since the transmitter amplifier is set to its lowest possible setting and, more importantly, we observe *even* harmonics of the transmitted tone, we can safely conclude that these harmonics must (at least partially) occur in the analog baseband signal. The only components that can introduce non-linearities in the transmitter side analog baseband signal are the two DACs. On the receiver side, we have two ADCs, which can also introduce non-linearities in the observed digital baseband signal. However, the ADC used in the NI 5791R transceiver [31] has a higher spurious-free dynamic range (SFDR) than the DAC [38], so we assume that all baseband non-linearities stem from the DACs.

We model the DAC-induced non-linearities by using a Taylor series expansion around 0 of maximum degree *m*
_{max}. In the block diagram of Figure 3, we see that the first DAC has ℜ{*x*} as its input and the second DAC has *I*{*x*} as its input. Thus the output signal of each DAC can be written as:

$$\begin{array}{*{20}l} \Re\{{\tilde{x}}\} & = \sum_{m=1}^{m_{\max}} \alpha_{1,m} \Re\{x\}^{m}, \end{array} $$

((4))

$$\begin{array}{*{20}l} \Im\{{\tilde{x}}\} & = \sum_{m=1}^{m_{\max}} \alpha_{2,m} \Im\{x\}^{m}, \end{array} $$

((5))

where \(\alpha _{i,m} \in \mathbb {R},~i\in \{1,2\},~m\in \{1,\ldots,m_{\max }\}\). Thus, the continuous time complex baseband signal \({\tilde {x}}\) can be written as:

$$\begin{array}{*{20}l} {\tilde{x}} = \sum_{m=1}^{m_{\max}} \alpha_{1,m}\Re\{x\}^{m} + j \sum_{m=1}^{m_{\max}} \alpha_{2,m} \Im\{x\}^{m}. \end{array} $$

((6))

By analysing Equation 6 with a single input tone of frequency *f*, it can be shown that, if the DACs are perfectly matched so that *α*
_{1,m
}=*α*
_{2,m
},*m*∈{1,…,*m*
_{max}}, the DAC induced non-linearities produce harmonics alternating on only one side of the spectrum for odd *m*, but on both sides of spectrum for even *m*. More specifically, it is shown in the appendix that for odd *m*, we obtain harmonics at frequencies \(m(-1)^{\frac {m-1}{2}}f\), while for even *m*, we obtain harmonics at both −*m*
*f* and *mf* with equal power. We observe in Figure 4b that the frequency 3*f* is not present but the frequency −3*f*, indicated by (iii), is present and also that the harmonics at −2*f* and 2*f* (indicated by (iv) and (v), respectively) have approximately equal power. Thus, all our observations are in complete agreement with what we expect to see based on our model and the proof in the appendix. Tone (ii), at frequency −*f*, which we observe clearly in Figure 4b but is not predicted by the DAC non-linearities, is the result of IQ imbalance, as we will explain in the following section.

It is interesting to note in Figure 4b that tone (iii), at −3*f*, is stronger than tone (ii), at −2*f*, which seems counterintuitive at first. However, when downscaling the digital baseband signal, we observe that the power of the third harmonic decreases at a higher rate than the power of the second harmonic, which is consistent with what one would expect.

### 4.3 IQ imbalance

IQ imbalance is caused by amplitude and phase mismatch in the in-phase and the quadrature components of the upconverted analog signal. To simplify notation, in this section, we consider frequency-flat IQ imbalance. The output of the non-ideal mixer can be modelled as:

$$\begin{array}{*{20}l} {{x}} & = \Re\left\{\left(\gamma_{\text{Tx}} {\tilde{x}} + \delta_{\text{Tx}} {\tilde{x}}^{*}\right) e^{j2\pi f_{c}t}\right\}, \end{array} $$

((7))

where \(\gamma _{\text {Tx}}, \delta _{\text {Tx}} \in \mathbb {C}\). We note that any amplitude mismatch in the linear components of the DACs will also manifest itself as IQ imbalance.

In Figure 4a, we observe that there exists a mirror image of the transmitted tone, with respect to the carrier frequency at frequency −*f* (indicated by (ii)) which arises due to the effect of IQ imbalance. However, it is important to note that the signal components that - at first sight - appear to be harmonics of this negative tone instead can only arise due to the DAC non-linearities as explained earlier. This is for several reasons: first, the harmonic of the original tone *f* at frequency 3*f* (not indicated in the figure as it lies below the thermal noise floor) is significantly weaker than the alleged harmonic of −*f* at frequency −3*f* (indicated by (iii)). Moreover, since there are no significant baseband non-linearities after the mixer of the receiver, we do not expect to observe a second harmonic of −*f* at frequency −2*f*. Finally, when we enable the built-in IQ imbalance compensation block of the NI 5791R transceivers, as demonstrated in Figure 4c, we observe that, while the power of the IQ imbalance induced tone (ii), at −*f*, is reduced by approximately 20 dB, the apparent harmonics of this tone at −2*f* and −3*f*, indicated by (iii) and (iv), respectively, are unaffected. In order to have IQ imbalance that is similar to what a low-cost transceiver would experience, we keep the built-in IQ imbalance compensation mechanism of the NI 5791R disabled.

### 4.4 RF non-linearities

Non-linearities in the upconverted RF signal are caused by the power amplifier that comes after the RF mixer. These non-linearities mainly appear when the amplifier is operated in its non-linear region, i.e., close to its maximum output power, where significant compression of the output signal occurs. Basic arithmetic manipulations can show that all the even-power harmonics lie out of band and will be cutoff by the RF low-pass filter of the receiver. The RF non-linearities can be modelled using a Taylor series expansion around 0 of maximum degree *n*
_{max}:

$$\begin{array}{*{20}l} {\hat{x}} & = \sum_{\substack{n=1,\\ n~\text{odd}}}^{n_{\max}}\beta_{n}{{x}}^{n}, \end{array} $$

((8))

where \(\beta _{n} \in \mathbb {R},~n\in \{1,3,\ldots,n_{\max }\}\).

The effect of RF non-linearities can be clearly seen in Figure 4d, where we present the spectrum of the received self-interference signal when transmitting with an output power of 20 dBm. We observe that strong third and fifth harmonics of the transmitted tone *f* appear (indicated by (vi) and (vii), respectively). The power of the tones on the negative frequencies with respect to the main tone *f* remains almost unaffected, as expected because they do not arise from the RF non-linearities, but from the DAC non-linearities. The third and fifth harmonics of the tones on the negative frequencies lie below the noise floor. We also observe that the noise floor has increased by 20 dB. This is caused by the effect of the limited dynamic range of the ADCs of the receiver, which means that quantisation noise dominates thermal noise. As can be seen by referring to the power budget in Figure 1, we would require at least 20 dB more passive or active analog suppression in order to observe the thermal noise floor.