Baseband and RF hardware impairments in full-duplex wireless systems: experimental characterisation and suppression

2.1 Active analog suppression

One of the most widely considered and successful active analog suppression techniques is to subtract a cancellation signal from the received signal (which contains the self-interference and any desired signal) [1,4,5]. A number of techniques for generating this cancellation signal have been proposed; however, these generally fall into two categories.

The Stanford architecture [4,32] proposes measuring the transmitted RF signal (e.g., with a low insertion loss coupler) immediately before it is applied to the antenna. The cancellation signal is generated by appropriately delaying and attenuating this measured signal to account for propagation in the real self-interference channel. A practical implementation of this approach using tapped delay lines achieved 45 dB active suppression, over a 40 MHz bandwidth at 2.45 GHz [4]. An advantage of the Stanford architecture is that hardware imperfections introduced by the transmitter circuit are inherently included in the cancellation signal. However, system-specific RF cancellation circuits are required to model the effect of the channel and the architecture does not scale well to multiple-input multiple-output (MIMO) systems, as each transmitter-receiver pair requires a dedicated circuit [32].

In contrast, the analog cancellation signal in the Rice architecture is generated using a separate RF chain [5,33]. The baseband input to the cancellation chain is computed by sounding the self-interference channel to determine the appropriate delay, attenuation, and distortion parameters that must be applied. Several implementations of the Rice architecture have been reported [5,34]. The initial prototype achieved between 20 and 34 dB analog suppression for transmit powers ranging from 0 and 15 dBm (generally suppression decreases with increasing transmit power), over a 40 MHz bandwidth on a WARP platform [5]. Similar results were reported using a National Instruments (NI) FlexRIO platform, with an average 48 dB active suppression over a 20 MHz bandwidth for 4 dBm transmit power [34]. For both implementations, further suppression was achieved in the digital domain; however, both groups observed the suppression introduced by digital cancellation depended on the analog stage and that the total suppression achieved was approximately constant [5,18,34]. The Rice architecture can be more readily extended to MIMO systems (as each receiver only requires one additional RF cancellation chain) and does not require specialised RF hardware design. However, it is important to note that hardware imperfections are potentially more prevalent in the Rice architecture, as the cancellation signal is generated from a separate RF chain.

2.2 Active digital suppression

The active analog suppression stage, which is usually not perfect, can be followed by suppression in the digital domain to further remove remaining self-interference. In general, i.e., with or without active analog suppression, the received complex baseband signal, r, can be expressed as a sum of: the external signal-of-interest, s; a function, f(·) of the complex baseband transmitted self-interference signal, x; and noise, z. The goal of digital suppression is to estimate f(·) and subtract f(x) from the received signal.

Since the transmitted signal is distorted by transmitter non-idealities, the self-interference signal is a complicated function of x. Sophisticated digital cancellation methods are required to capture those transmitter imperfections [24,25]. A thorough identification and characterisation of these non-idealities is presented in Section 4, and digital cancellation schemes are presented in Section 5.

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.

where LPF denotes a low-pass filter that removes the copies of the signal around 2f _c and −2f _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.

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 3f is not present but the frequency −3f, indicated by (iii), is present and also that the harmonics at −2f and 2f (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 −3f, is stronger than tone (ii), at −2f, 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

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 3f (not indicated in the figure as it lies below the thermal noise floor) is significantly weaker than the alleged harmonic of −f at frequency −3f (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 −2f. 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 −2f and −3f, 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

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.

5.1 Existing digital cancellation methods

where f(x) denotes a function of the complex baseband self-interference signal x, s denotes the signal-of-interest and z denotes thermal noise. The goal of digital suppression is to estimate the function f and subtract f(x) from the received signal. In order to do so, f needs to be modelled in some way.

where h _SI,n , n=1,3,…,n _max, denotes the self-interference channel experienced by each of the harmonics of the baseband signal. In this case, n _max channels are estimated using the LS method, in order to remove the baseband signal and harmonics thereof from the received signal.

where h _SI and h _SI,IQ denote the channels experienced by the baseband signal and the complex conjugate of the baseband signal, respectively. The goal becomes to jointly estimate h _SI and h _SI,IQ using the LS method.

5.2 Joint digital cancellation of DAC non-linearities and IQ imbalance

Ideally, we would like to perform digital cancellation based on a model that includes all non-idealities. However, due to multiple non-linearities, a full model is highly complicated. Thus, we first examine the case where the output power is low, so that it can be safely assumed that we are only ignoring very limited RF non-linearities. In this case, the resulting non-idealities model leads to a convenient cancellation method.

5.2.2 5.2.2 Performance of digital cancellation

In order to assess the performance of different digital cancellation mechanisms on our testbed, we conduct the following experiment. We construct 100 frames containing a 10 MHz 4-QAM modulated signal. We transmit each group of frames using transmit powers ranging from −10 up to 22 dBm. The carrier frequency is set to 2.48 GHz. We consider two antenna spacings that give us 40 and 55 dB of passive analog suppression. In practice, these suppression numbers are easily achievable by using a combination of passive suppression and active analog suppression, but as mentioned earlier, in our experiments, no active analog cancellation is performed to keep all signal components accessible. The digital baseband samples are recorded, and the various digital cancellation methods are applied to them off-line.

In Figure 5a, we present the mean distance of the residual self-interference signal from the measured noise floor for several digital cancellation methods with 40 dB of passive suppression. Since the amount of cancellation is a random variable, we also include error bars at one standard deviation from the mean. We also present the apparent noise floor at each transmit power, which stems mainly from the limited dynamic range of the ADC of the receiver. The achievable cancellation of any cancellation method is limited by this apparent noise floor, and the only way to overcome it is to increase the amount of cancellation in the analog domain. We observe that simple linear cancellation alone is insufficient even at low transmit powers, as the residual signal lies at least 7 dB above the noise floor. Non-linear cancellation [24] only reduces the residual self-interference by an additional 2 dB over the entire examined range of transmit powers. The IQ imbalance cancellation method proposed in [25] improves the obtained cancellation drastically. However, our joint cancellation method consistently outperforms all previously proposed methods as it considers both the baseband non-linearities and the IQ imbalance.

In Figure 5b, we present the mean distance of the residual self-interference signal from the measured noise floor for several digital cancellation methods with 55 dB of passive suppression. In this case, the apparent noise floor remains equal to the measured thermal noise floor for all considered transmit powers. Moreover, we observe that at low transmit powers, almost all cancellation methods perform equally well because the required amount of suppression is relatively low and most of the non-idealities lie below the thermal noise floor. As the transmit power is increased, however, IQ imbalance cancellation and our joint cancellation perform significantly better than linear and non-linear cancellation. It is important to note that our joint cancellation method manages to keep the residual self-interference at less than 3 dB above the noise floor up to a transmit power of 18 dBm. However, all cancellation methods start failing at transmit powers above 18 dBm.

7.1 DAC non-linearities

Thus, when s _k,m =+1 all cos(ω _c t+ω _m,k t) terms disappear. On the other hand, when s _k,m =−1, all cos(ω _c t−ω _m,k t) terms disappear. Thus, with ideal downconversion, the harmonics resulting from odd values of m appear at frequencies \(m(-1)^{\frac {m-1}{2}}\omega \) in the analog baseband.

By replacing Equations 32 and 33 in 26, it can be shown similarly that no terms cancel out. Thus, with ideal downconversion, the harmonics resulting from even values of m appear at both −m ω and +m ω in the analog baseband signal.