Subspace-based self-interference cancellation for full-duplex MIMO transceivers

Consider two transceivers communicating in a full-duplex fashion. The simultaneous transmission and reception creates self-interference (SI) to be cancelled before the demodulation process. The SI signal is first suppressed at RF, prior to the low-noise amplifier (LNA) and analog-to-digital converter (ADC) to avoid overloading/saturation of these components [2, 3, 32]. In [5], we proposed an efficient compressed-sensing (CS)-based algorithm for the RF SI cancellation stage. In this work, we concentrate on the development of subspace-based algorithm to jointly estimate the SI and intended channels and the nonlinear distortions for the baseband SI cancellation stage of a full-duplex MIMO transceiver with arbitrary numbers of transmit and receive antennas. The output signal of the RF SI cancellation stage consists of the residual SI, the intended signal received from the other transceiver and the additive thermal noise. Figure 1 shows a simplified block diagram of a MIMO transceiver. The residual SI can be further suppressed at the baseband after ADC using digital signal processing (DSP). The advantage of working in the digital domain, as compared to RF, is that sophisticated DSP methods can be handled. Both transceivers are equipped with N _t transmitting antennas and N _r receiving antennas. At transmitting antenna q, a group of N data symbols X _q =[ X _q (0),…, X _q (N−1)] ^T is first modulated by the IFFT matrix to form an OFDM block, then the time domain vector x _q =[ x _q (0),…, x _q (N−1)] ^T is extended by the cyclic prefix of length¹ N _cp and the resulting vector is sent sequentially. In the transmit stream q, the complex signal x _q (t) after the digital-to-analog conversion (DAC), is passed through an imbalance IQ mixer whose output is as follows:

$$ x_{q}^{IQ}(t) = k_{1,q} x_{q}(t) + k_{2,q} x_{q}^{*}(t), $$

(1)

For multi-block transmission, the received vector in (14) is indexed by the block number t, i.e., y _t . For convenience, we omit this indexation and we will consider later a given number of transmitted blocks to compute the covariance matrix of the received vector.

as long as the signal samples are uncorrelated from the noise samples².

It is worth mentioning that the proper noise has a vanishing pseudo-covariance [34]. The main purpose of using the extended received signal is to increase the dimension of the received signal and thus avoid the degenerate noise subspace. Hence, the subspace identification procedure can be derived only if the signal part covariance matrix, given by \(\widetilde {\boldsymbol {H}} \boldsymbol {R}_{\widetilde u} \widetilde {\boldsymbol {H}}^{H}\), of the covariance matrix \(\boldsymbol {R}_{\widetilde y}\) is singular. It results that \(d_{s} = \text {rank}(\widetilde {\boldsymbol {H}} \boldsymbol {R}_{\widetilde u} \widetilde {\boldsymbol {H}}^{H}) < 2MN_{r}\). In this case, the signal is confined in a d _s -dimensional subspace and the remaining noise subspace is with dimension 2MN _r −d _s . Singularity of \(\boldsymbol {R}_{\widetilde u}\) is a necessary condition to obtain a nondegenerate noise subspace. Actually, noting that \(\widetilde {\boldsymbol {H}}\) is full rank, nonsingular \(\boldsymbol {R}_{\widetilde u}\) results in \(\text {rank}(\widetilde {\boldsymbol {H}} \boldsymbol {R}_{\widetilde u} \widetilde {\boldsymbol {H}}^{H}) = 2MN_{r}\), and thus, the matrix \(\widetilde {\boldsymbol {H}} \boldsymbol {R}_{\widetilde u} \widetilde {\boldsymbol {H}}^{H}\) spans all the observation space. On the other hand, since the matrix \(\widetilde {\boldsymbol {H}}\) is a tall matrix, singularity of \(\boldsymbol {R}_{\widetilde u}\) is not a sufficient condition to guarantee the singularity of \(\widetilde {\boldsymbol {H}} \boldsymbol {R}_{\widetilde u} \widetilde {\boldsymbol {H}}^{H}\).

In the following, we distinguish two cases of real and complex modulated symbols.

For real modulated symbols, it can be shown that \(\boldsymbol {R}_{\widetilde u} = \alpha ^{2} \boldsymbol {M} \otimes \boldsymbol {I}_{2N_{t}}\) with the 2N×2N matrix M having the following form:

(22)

From (22), we note that each column of M appears exactly two times (the first column of M is the same as the (N+1) ^th column, and the i ^th column of M is the same as the (2N−i+2) ^th column, for i=2,…, N). Therefore, the matrix M has exactly N-independent columns and thus its rank is N. It follows that the rank of \(\boldsymbol {R}_{\widetilde u}\) is 2NN _t . In Appendix 1, we show that \(\boldsymbol {R}_{\widetilde u}\) has zero eigenvalue with multiplicity 2NN _t and 2α ² also with multiplicity 2NN _t . Then, the matrix \(\boldsymbol {R}_{\widetilde u}\) is decomposed as U D U ^H where D is the 4NN _t ×4NN _t diagonal matrix with zeroes in the first 2NN _t diagonal elements and 2α ² in the last 2NN _t diagonal elements and U is an orthogonal matrix whose columns are the corresponding eigenvectors of \(\boldsymbol {R}_{\widetilde u}\).

where P and Q are two matrices. By combining the data symbol X _q and its complex conjugate, we force the pseudo-covariance matrix to be different from zero. Appendix 2 gives a detailed discussion about the choice of the matrices P and Q so that the covariance matrix \(\boldsymbol {R}_{\widetilde u}\) has rank 2NN _t and can be decomposed as U D U ^H with D as the 4NN _t ×4NN _t diagonal matrix with zeroes in the first 2NN _t diagonal elements.

and j is the complex number satisfying j ²=−1.

We mention that the matrices U ₂ and U ₄ do not depend on the received signal and can be computed offline prior to the transmission. It is also seen that the overestimated channel order L does not affect the estimation process. This is a common property with other subspace-based estimators [17].

and \(\boldsymbol {\check H}^{(s)} = [{\boldsymbol {H}^{(s)}}^{T}(0),~{\boldsymbol {H}^{(s)}}^{T}(1),\dots,~{\boldsymbol {H}^{(s)}}^{T}(L)]^{T}\) can be also written as \(\boldsymbol {\check H}^{(s)} = \boldsymbol {\check \Phi }_{s} (\boldsymbol {I}_{N_{t}} \otimes \boldsymbol {c}),\) where \(\boldsymbol {\check \Phi }_{s}\) is defined in the same way as \(\boldsymbol {\check \Phi }_{i}\). \(\boldsymbol {\check H}^{(i)}\) and \(\boldsymbol {\check \Phi }_{i}\) are used to build the matrices H ⁽ⁱ⁾ and Ψ _i , respectively, having the same block structure as H in (13).

where, for clarity, we introduce \(\phantom {\dot {i}\!}\boldsymbol {B} = \boldsymbol {1}_{N} \otimes \boldsymbol {I}_{N_{t}}\) and \(\phantom {\dot {i}\!}\widehat {\boldsymbol {C}}_{k-1} = \boldsymbol {I}_{NN_{t}} \otimes \widehat {\boldsymbol {c}}_{k-1}\) and we use the equality \(\Big (\big ((\boldsymbol {I}_{N} \otimes \boldsymbol {K})\boldsymbol {x}_{i}^{*} \big) \otimes \boldsymbol {I}_{4N_{t}N_{r}} \Big) \boldsymbol {c} = \Big (\big (\text {diag}(\boldsymbol {x}_{i}^{*})\boldsymbol {B} \big) \otimes \boldsymbol {c} \Big) \boldsymbol {k}\). Then, \(\widehat {\boldsymbol {s}}_{k}\) is transformed in the frequency domain and each element of the frequency domain vector is projected to its closest discrete constellation point. The obtained vector is converted back to the time domain to obtain a better estimate \(\widetilde {\boldsymbol {s}}_{k}\) of s.

In this section, we provide some simulation results on the performance of the proposed estimation algorithm for a 2×2 MIMO full-duplex system. The transmitted bits are mapped to 4-QAM symbols, then passed through an OFDM modulator of length N=64. The wireless channel is represented as a Rayleigh multipath fading channel with five equal-variance resolvable paths. Since the exact number of paths is supposed to be unknown, the algorithm is parametrized as if there are eight paths. In the following, the SNR is defined as the average intended-signal-to-thermal noise power ratio and the estimation mean square error (MSE) of H is \(\textrm {MSE} = E\Big (||\boldsymbol {H} - \widehat {\boldsymbol {H}}||^{2}\Big).\) To model the RF impairments, a complete transmission chain is simulated. The PA coefficients are derived from the intercept points by taking the IIP 3=20 dBm. For the IQ mixer, the ratio between the direct signal and the image is set to 28 dB which is specified in 3GPP LTE specifications [23]. The ADC is modelled as a 14-bit quantizer to incorporate the quantization noise. Therefore, no simplifications are made regarding the different impairments. Antenna separation can attenuate the SI by 40 dB while the RF cancellation stage reduces the direct path by 30 dB [1] leaving the weaker reflections and transceiver impairments to be reduced by the proposed digital algorithm.

where \(\boldsymbol {R} = \alpha ^{2} {\boldsymbol {H}^{(s)}}^{H}\boldsymbol {H}^{(s)} + \sigma ^{2} \boldsymbol {I}_{N_{r}M}\). An iterative procedure to find the ML estimate was proposed in [35]. The covariance matrix is obtained by averaging 60 OFDM blocks. Figures 2 and 3 plot the MSE vs. SNR curves for the SI and intended channel estimations, respectively. In both figures, one pilot symbol, from the intended transceiver, is used to solve the ambiguity matrix. For comparison purpose, a perfect estimate of the ambiguity term c is obtained as \(\boldsymbol {c}_{perfect} = \arg \min _{\boldsymbol {c}}||\boldsymbol {\check h} - \boldsymbol {\Phi }\boldsymbol {c}||_{2}^{2}\) and the corresponding curves are labelled by clairvoyant subspace. It is seen that, when one pilot symbol is used in the ML and LS estimators, the proposed subspace algorithm offers notably lower MSE over a large SNR range. We also represent the performance of the ML and LS estimators when 20% of the transmit symbols are known (pilot symbols equally spaced within one OFDM symbol) while keeping one pilot symbol for the subspace method⁴. In this case, the three algorithms give comparable performance at low SNR region with the expanse of lower bandwidth efficiency. As the SNR increases, the performance of the LS and ML estimators saturate due to the reduced number of pilot symbols and the presence of the unknown transmit signal from the intended transceiver which acts as an additive noise. While the subspace algorithm exploits the information bearing in the unknown data to find the signal subspace. The ambiguity term is first solved using the known transmit symbols, then the iterative decoding ambiguity estimation is applied to improve the estimation performance. From Figs. 2 and 3, three to four iterations are sufficient to converge and the performance is close to the performance when the ambiguity term c is perfectly obtained. Note that the ML solution is also obtained in an iterative way and for a fair comparison; we simulate the performance of the ML estimator after four iterations. As it can be expected, the estimate of the SI channel is more accurate than the estimate of the intended channel. This can be explained by the fact that the self-signal is known while one pilot symbol is known in the intended signal.

The number of pilot symbols is a critical issue in channel estimation since a large pilot sequence provides better estimation performance but reduces the bandwidth efficiency of the system. In Figs. 4 and 5, we compare the impact of the number of pilot symbols on the performance of the three estimators. We periodically place the pilot symbols within an OFDM symbol. Optimal pilot placement requires to verify all P _pilot combinations from N subcarriers and hence, leads to an NP-hard problem beyond the scope of this paper, and is left for future work. It can be seen from these figures that the subspace method is not greatly affected by the number of pilot symbols since the subspaces are obtained using the second-order statistics of the received signal and not the transmit signal itself. Clearly, the proposed algorithm outperforms the ML and LS estimators at a reduced number of pilots while this tendency is inverted when the number of pilots increases. However, a system with a large amount of pilot symbols is not of practical interest.

In Figs. 6 and 7, we evaluate the impact of the number of observed OFDM symbols on the estimation performance. For the three algorithms, we consider the transmission scheme where the number of pilot symbols is set to one and the SNR is 10 dB. As the subspace algorithm is based on estimates of the second-order statistic of the received signal, its performance varies with the number of OFDM symbols. All three algorithms are able to estimate the SI channel with an error floor for the LS. The ML and subspace algorithms offer the similar performance. On the other hand, the LS estimator fails to recover the intended channel, for any number of OFDM symbols. This can be explained by the fact that the number of unknowns (intended channel coefficients) is larger than the number of pilot symbols. Hence, it is not possible to use this method when the number of pilot symbols is small. The ML estimator presents also poor estimation performance for the intended channel, while the subspace method is able to return a good channel estimate, with a better bandwidth efficiency compared to the other estimators, as soon as there are enough OFDM symbols to compute the covariance matrix.

Our primary motivation of this work is to develop an accurate channel estimator to cancel the SI signal. The performance of the SI-canceller are represented by its achieved output signal-to-residual-SI-and-noise power ratio (SINR) after SI cancellation vs. the input SNR. Ideally, if SI could be completely cancelled then the residual SI after cancellation is 0, and consequently, the output SINR equals the input SNR as shown by the dashed line “perfect cancellation” in Fig. 8. In other words, the “perfect cancellation” is considered as the ideal upper-bound for the SINR. As shown in Fig. 8, with three iterations, the proposed subspace-based SI-canceller can offer an output SINR very close to the upper-bound over a large SNR range. At low SNR, the large estimation error results in a larger residual SI after cancellation, which ultimately affects the output SINR.

We also investigate in Fig. 8 a frequency domain method to estimate the different parameter using the pilot symbols on some subcarriers. We resort to the LS estimator to find the channel responses at the pilot subcarriers. Since the remaining subcarriers contain unknown symbols from the intended transceiver, the complete channel responses are obtained by linear interpolation of the estimated coefficients. Thus, the frequency domain approach uses only the portion of the signal containing pilots while the proposed approach exploits the whole received signal through the second-order statistics. Clearly, the performance of the frequency domain approach highly depends on the number of pilots (as shown in Fig. 8) since the interpolation cannot model the variance of the channel in the frequency domain. We also compare the proposed method with the widely linear LS estimator in [26]. Note that the algorithm in [26] ignores the PA nonlinearities and does not incorporate the intended signal in the estimation process. Some time frames are dedicated to transmit orthogonal pilot symbols for estimation purpose, where the transceiver receives only its own signal. Therefore, the widely linear LS estimator incurs an overhead and requires synchronization between the two transceivers. Besides, it shows a noise floor at high SNR because the PA nonlinearity is not considered during the estimation process. On the other hand, by exploiting the whole received signal through its second-order statistics, the proposed method offers good performance even with one pilot and still outperforms the frequency domain approach (even with much larger number of pilots). Figure 9 plots the bit error rate (BER) vs. SNR curves of the two approaches. For comparison, we include the case of perfect channel estimate. To improve the BER, the SINR should be kept as high as possible at the demodulator. To conclude, while the frequency domain approach is more intuitive, it needs a large number of pilots and is outperformed by the proposed method.

We evaluate the performance of the system in the presence of phase noise by simulation. Figures 10 and 11 plot respectively the SINR and the BER vs. the phase noise 3 dB bandwidth f _3dB for SNR =20 dB and common oscillator at the transmitter and the receiver. The residual SI depends obviously on the quality of the oscillator represented by its f _3dB . Higher f _3dB results in a fast varying process. Clearly, the proposed method still offers good cancellation performance, which is degraded as f _3dB increases.

The PA nonlinearity effects on the performance of the proposed algorithm are also investigated through simulations. Figure 12 plots the resulting SINR after cancellation vs. the value of the PA third-order intercept point (IIP3) for SNR =20 dB. For perfect cancellation, the resulting SINR after cancellation would be the SNR =20 dB. A lower IIP3 indicates higher PA distortions (or poorer PA) and hence reduces the resulting SINR after cancellation. Figure 12 shows that as the IIP3 value increases, the cancellation performance is improved. However, for a sufficiently high IIP3 (e.g., 18 dBm or higher), the PA distortions are no longer dominant and the resulting SINR after cancellation is unchanged. This can be explained by the fact that, when developing the algorithm, the third-order component of the signal \(x_{q,ip3}(n) = x_{q}^{IQ}(n)|x_{q}^{IQ}(n)|^{2}\) is approximated by x _q (n)|x _q (n)|² to simplify the algorithm. This approximation only affects the algorithm performance when the nonlinear coefficients are sufficiently high.

¹ The length of the cyclic prefix N _cp should be larger than the delay spread of the channel to eliminate the inter-symbol interference and inter-carrier interference. Therefore, if we know the length of the channel, we can set the cyclic prefix to be sufficiently large to satisfy N _cp >L. Since this information is in general not available, N _cp is chosen to guarantee N _cp >L. For example, if the distance between the two transceivers is 1 km, a cyclic prefix of 4 microsec is sufficient.

² Physically, the additive noise arises from the thermal agitation of the charge carriers in an electronic device and is independent from the input. It can also contain interference from other systems whose signals are independent from the transmit signal of the considered system.

³ The previous condition is verified for independent channels between different antennas.

⁴ The pilot symbols are equally spaced within one OFDM symbol.

First, if τ=1 is an eigenvalue of M, then it exists a vector a≠0 such that M a−a=0. It follows that a(1)=a(2)=⋯=a(2N)=0, which is in contradiction with a≠0. Therefore, 1 is not an eigenvalue of M.

where we used the fact that M _1,2 M _1,2=I _N . Then, the solutions to det(M−τ I _2N )=0 are 0 and 2. Therefore, all non-zero eigenvalues of M are equal to 2 and thus all the non-zero eigenvalues of \(\boldsymbol {R}_{\widetilde u}\) are equal to 2α ².