SCIENCE CHINA Information Sciences, Volume 60 , Issue 10 : 102302(2017) https://doi.org/10.1007/s11432-016-0291-9

## Novel multi-tap analog self-interference cancellation architecture with shared phase-shifter for full-duplex communications

• AcceptedSep 30, 2016
• PublishedMar 28, 2017
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### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61531009, 61501093, 61271164, 61471108) and Fundamental Research Funds for the Central Universities.

### Supplement

Appendix

SIC performance provided by CMTS

Replacing $\boldsymbol{A}$ in (5) by $\boldsymbol{A}_\text{CMTS}=[a_1\exp(\text{j} \phi_1)a_2\exp(\text{j} \phi_2)\cdots$ $a_N\exp(\text{j} \phi_N)]^\text{T}$ yields the power of residual SI of CMTS $P_\text{e,CMTS}$, where $\phi_1,\phi_2,\dots,\phi_N$ are the phase offsets of the $N$ variable phase shifters in CMTS. The optimal $\boldsymbol{A}_\text{CMTS}$ to minimize $P_\text{e,CMTS}$ is an unconstrained minimization problem and can be solved by ordering $0=\nabla P_\text{e,CMTS}$[19], i.e., \left\{\begin{aligned}&0={\partial P_\text{e,CMTS}}/{\partial\textrm{Re}\{\boldsymbol{A}_\text{CMTS}\}} =-2\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}\} +2\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\textrm{Re}\{\boldsymbol{A}_\text{CMTS}\} -2\textrm{Im}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\textrm{Im}\{\boldsymbol{A}_\text{CMTS}\}, \\ &0={\partial P_\text{e,CMTS}}/{\partial\textrm{Im}\{\boldsymbol{A}_\text{CMTS}\}} =-2\textrm{Im}\{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}\} +2\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\textrm{Im}\{\boldsymbol{A}_\text{CMTS}\} +2\textrm{Im}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\textrm{Re}\{\boldsymbol{A}_\text{CMTS}\}. \end{aligned}\right. \tag{18} The solution of (18) is derived as $\boldsymbol{A}_\text{CMTS}=\boldsymbol{O}^{-1}\boldsymbol{R}_{b}^{-1}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}$, which is the well-known Wiener solution 3). Then the SIC performance provided by CMTS is computed with (sect. 3.3) and given as $$G_\text{CMTS}=I_{\text{t/r}}(I_{\text{t/r}}-\boldsymbol{H}^\text{H}\boldsymbol{Q}^\text{H}\boldsymbol{C}^\text{H}_b\boldsymbol{R}_{b}^{-1}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H})^{-1}. \tag{19}$$

Diniz P S R. Adaptive filtering—algorithms and practical implementation. 3rd ed. Spring Street, NY: Springer Science & Business Media, 2008. 25–47.

SIC performance provided by DMTS

In DMTS, the power of residual SI $P_\text{e,DMTS}$ is obtained by replacing $\boldsymbol{A}$ by $\boldsymbol{\tilde{A}}$ in (5). Finding the optimal tap coefficients in DMTS to minimize $P_\text{e,DMTS}$ is expressed as \begin{aligned}\min &P_\text{e,DMTS} \\ \textrm{subject to} &\boldsymbol{\tilde{A}}\ge0\end{aligned} \Leftrightarrow \ \begin{aligned}\min \Bigg\|&\left[ \begin{matrix} \operatorname{Re}( {{\boldsymbol{\Lambda }}^{1/2}}{{\boldsymbol{U}}^{-1}}\boldsymbol{O} ) \\ \operatorname{Im}( {{\boldsymbol{\Lambda }}^{1/2}}{{\boldsymbol{U}}^{-1}}\boldsymbol{O} ) \\ \end{matrix} \right]\boldsymbol{\tilde{A}}-\left[ \begin{matrix} \operatorname{Re}( {{\boldsymbol{\Lambda }}^{-1/2}}{{\boldsymbol{U}}^{-1}}{{\boldsymbol{C}}_{b}}\boldsymbol{QH} ) \\ \operatorname{Im}( {{\boldsymbol{\Lambda }}^{-1/2}}{{\boldsymbol{U}}^{-1}}{{\boldsymbol{C}}_{b}}\boldsymbol{QH} ) \\ \end{matrix} \right] \Bigg\|_{2}^{2} \\ &\textrm{subject to} \boldsymbol{\tilde{A}}\ge0\end{aligned}, \\ \tag{20} where $\boldsymbol{R}_{b}=\boldsymbol{U}\boldsymbol{\Lambda}\boldsymbol{U}^\text{H}$$^{2)}, \boldsymbol{\Lambda} is the diagonal matrix consisting of the eigenvalues of \boldsymbol{R}_{b}, \boldsymbol{U} is the unitary matrix consisting of the normalized eigenvectors of \boldsymbol{R}_{b}, and \|\cdot\|_2 is the Euclidean norm of a vector[20]. Eq. (20) is an nonnegative least squares optimization problem and can be solved with various well-developed numerical approaches 4), and then the SIC performance provided by DMTS is computed with (sect. 3.3) and given as $$G_\text{DMTS}=I_{\text{t/r}}({{I}_\text{t/r}} -2\textrm{Re}\{\boldsymbol{H}^\text{H}\boldsymbol{Q}^\text{H}\boldsymbol{C}^\text{H}_b\boldsymbol{O}\}\boldsymbol{A}_\text{DMTS}+ \boldsymbol{A}_\text{DMTS}^\text{T}\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\boldsymbol{A}_\text{DMTS})^{-1}, \tag{21}$$ where \boldsymbol{A}_\text{DMTS} is the numerical result of (20). The upper bound of {G}_\text{DMTS} is derived as follows. Relaxing the constraint \boldsymbol{\tilde{A}}\ge0, the optimal \boldsymbol{\tilde{A}} for (20) is derived as \boldsymbol{\hat{A}}_\text{DMTS}=\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}^{-1} \textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}\} by ordering 0=\nabla P_\text{e,DMTS}[19], and then the corresponding SIC performance is computed with (5) and (sect. 3.3) as \hat{G}_\text{DMTS}=I_{\text{t/r}}({{I}_\text{t/r}} - \textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}\}^\text{T} \textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}^{-1} \textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}\})^{-1}. To compare \hat{G}_\text{DMTS} with {G}_\text{DMTS}, 1/{G}_\text{DMTS}-1/\hat{G}_\text{DMTS}=\|\boldsymbol{\Sigma}^{-1/2}\boldsymbol{V}^\text{H}\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}\} -\boldsymbol{\Sigma}^{1/2}\boldsymbol{V}^\text{H}\boldsymbol{A}_\text{DMTS}\|_2^2/I_{\text{t/r}}\ge0 is computed, and then \hat{G}_\text{DMTS}\ge {G}_\text{DMTS} is derived, i.e., \hat{G}_\text{DMTS} is the upper bound of {G}_\text{DMTS}, where \textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}=\boldsymbol{V}\boldsymbol{\Sigma}\boldsymbol{V}^\text{H}, \boldsymbol{\Sigma} is the diagonal matrix consisting of the eigenvalues of \textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}, \boldsymbol{V} is the unitary matrix consisting of the normalized eigenvectors of \textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}. Chen D, Plemmons R. Nonnegativity constraints in numerical analysis. In: Bultheel A, Cools R, eds. The Birth of Numerical Analysis. Hackensack: World Scientific, 2010. 109–139. Reconstruction power efficiencies of CMTS and DMTS After G_\text{CMTS} and G_\text{DMTS} are maximized, the reconstruction power efficiencies of CMTS and DMTS are computed by substituting \boldsymbol{A}_\text{CMTS} and \boldsymbol{A}_\text{DMTS} into (14) and given as $$\left\{\begin{array}{l} \eta_\text{CMTS}= ({(\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H})^\text{H}\boldsymbol{R}_{b}^{-1}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}})/({(\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H})^\text{H} \boldsymbol{R}_{b}^{-2}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}})/2^{\lceil\log_2(N)\rceil}, \\ \eta_\text{DMTS}=({\boldsymbol{A}_\text{DMTS}^\text{T}\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\boldsymbol{A}_\text{DMTS}})/({\boldsymbol{A}^\text{T}_\text{DMTS}\boldsymbol{A}_\text{DMTS}})/2^{\lceil\log_2(N)\rceil}, \end{array}\right. \tag{22}$$ respectively, where the fixed power combiner arrays in CMTS and DMTS are assumed to also have the tree structure, and thus likewise have an insertion loss 2^{\lceil\log_2(N)\rceil}. With the property of the Rayleigh quotient[20], \eta_\text{DMTS}\in[\lambda_\text{min}\{\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\}/2^{\lceil\log_2(N)\rceil}, \lambda_\text{max}\{\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\}/2^{\lceil\log_2(N)\rceil}] and \eta_\text{CMTS}\in[\lambda_\text{min}\{\boldsymbol{R}_b\}/2^{\lceil\log_2(N)\rceil}, \lambda_\text{max}\{\boldsymbol{R}_b\}$$/2^{\lceil\log_2(N)\rceil}]$ can be derived from (22). Combining with (15), the variation ranges of $\eta_\text{CMTS}$ and $\eta_\text{DMTS}$ are summarized as $$\left\{\begin{array}{l} \eta_\text{CMTS}\in[\lambda_\text{min}\{\boldsymbol{R}_{b}\}/2^{\lceil\log_2(N)\rceil}, \lambda_\text{max}\{\boldsymbol{R}_{b}\}/2^{\lceil\log_2(N)\rceil}], \\ \eta_\text{DMTS}\in[\max(\lambda_\text{min}\{\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\}, \lambda_\text{min}\{\boldsymbol{R}_b\})/2^{\lceil\log_2(N)\rceil}, \min(\lambda_\text{max}\{\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}\}, \lambda_\text{max}\{\boldsymbol{R}_b\})/2^{\lceil\log_2(N)\rceil}], \end{array}\right. \tag{23}$$ respectively.

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• Figure 1

Multi-tap analog SIC schemes. (a) CMTS; (b) DMTS.

• Figure 2

The proposed SMTS.

• Figure 3

Design of the reconfigurable power combiner array. (a) Structure of a SuDiC; (b) structure of the proposed reconfigurable power combiner array, where $x_i$ represents the signal from the $i$th tap.

• Figure 4

(a) Considered FD transceiver frontend; (b) power delay profile of SI channel.

• Figure 5

(Color online) Convergence of the developed numerical algorithm with 7 taps, i.e., $N=7$. (a) $G_\text{SMTS}$ vs. iteration time for 20-MHz SI; (b) $G_\text{SMTS}$ vs. iteration time for 100-MHz SI; (c) $\phi$ vs. iteration time for 20-MHz SI; (d) $\phi$ vs. iteration time for 100-MHz SI.

• Figure 6

(Color online) SIC performance vs. tap number. (a) 20-MHz SI; (b) 100-MHz SI.

• Figure 7

(Color online) CDF of SIC performance. (a) 20-MHz SI and $N=3$; (b) 20-MHz SI and $N=5$; (c) 20-MHz SI and $N=7$; (d) 100-MHz SI and $N=3$; (e) 100-MHz SI and $N=5$; (f) 100-MHz SI and $N=7$. Simulations are performed by 3000 times.

• Figure 8

(Color online) Coupling channel. The tap number is 7, i.e., $N=7$. (a) Magnitude response with 20-MHz SI; (b) magnitude response with 100-MHz SI; (c) time domain response with 20-MHz SI; (d) time domain response with 100-MHz SI.

• Figure 9

(Color online) CDF of reconstruction power efficiency. The delay interval of the delay lines is $\Delta\tau=4$ ns. (a) 20-MHz SI and $N=3$; (b) 20-MHz SI and $N=5$; (c) 20-MHz SI and $N=7$; (d) 100-MHz SI and $N=3$; (e) 100-MHz SI and $N=5$; (f) 100-MHz SI and $N=7$. Simulations are performed by 3000 times.

•

Algorithm 1 The numerical algorithm to solve (10)

Require:Given threshold $P_\text{th}$, one temporary variable $k$;

$\boldsymbol{\tilde{A}}_0\Leftarrow0$, $P_\text{e}(0)\Leftarrow {{{P}_{\text{tx}}}}I_{\text{t/r}}$, $\phi_0\Leftarrow0$, $k\Leftarrow0$;

repeat

$k\Leftarrow k+1$;

$\boldsymbol{\tilde{A}}_k\Leftarrow\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\}^{-1}\textrm{Re}\{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}\exp(-\text{j} \phi_{k-1})\}$;

$\phi_k\Leftarrow\text{Ang}\{{\boldsymbol{O}^\text{H}\boldsymbol{C}_b\boldsymbol{Q}\boldsymbol{H}^\text{T}\boldsymbol{\tilde{A}}_k}\}$;

$P_\text{e}(k)\Leftarrow{{{P}_{\text{tx}}}}(I_{\text{t/r}}-\textrm{Re}\{\boldsymbol{H}^\text{H}\boldsymbol{Q}^\text{H}\boldsymbol{C}^\text{H}_b\boldsymbol{O}\boldsymbol{\tilde{A}}_k\exp(\text{j} \phi_k)\}+\boldsymbol{\tilde{A}}_k^\text{T}\boldsymbol{O}^\text{H}\boldsymbol{R}_b\boldsymbol{O}\boldsymbol{\tilde{A}}_k)$;

until $\|P_\text{e}(k)-P_\text{e}(k-1)\|\le P_\text{th}$

• Table 1   Various multi-tap analog SIC prototypes
 Paper Type Number of taps BW (MHz) SI reduction$^\text{a)}$ (dB) Tuning time$^\text{b)}$ (ms) Ref. [5] CMTS 4 20 43 10 Ref. [6] CMTS 4 30 31 – Ref. [7] CMTS 10 20 57 – Ref. [8] CMTS 4 20 35 – Ref. [9] DMTS 2 20 13 – Ref. [10] DMTS 2 10 20 0.13 Ref. [11] DMTS 16 20 57 1 Ref. [12] DMTS 12 20 53 –
• Table 2   Comparisons of CMTS, DMTS, and SMTS with $N$ taps
 Item CMTS DMTS SMTS 1. Number of fixed delay lines $N$ $N$ $N$ 2. Number of variable scalars $N$ $N$ $N$ 3. Number of variable phase shifters $N$ 0 1 4. $N$-way power combiner array 1 1 1 5. Dimensions of control algorithm $2N$ $N$ $N+1$
• Table 3   Comparisons of CMTS, DMTS, and SMTS with $N$ taps
 Item CMTS DMTS SMTS 1. SIC performance Highest Lowest Medium 2. Quantity of variable phase shifters $N$ 0 $1$ 3. Quantity of power combiners$^\text{a)}$ $2^{\lceil\log_2(N)\rceil}-1$ $2^{\lceil\log_2(N)\rceil}-1$ $2^{\lceil\log_2(N)\rceil}-1$ 4. Quantity of selectors 0 0 $2^{\lceil\log_2(N)\rceil}-1$ 5. Reconstruction power efficiency Lowest Medium Highest 6. Dimensions of control algorithm $2N$ $N$ $N+1$

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