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Inner and outer bounding capacity region for multiple-access relay channel with non-causal side information at one encoder

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Abstract

In this paper, we analyze inner and outer bounding capacity region of two-user multiple-access relay channel (MARC) with side information (SI) non-causally known at one encoder in different scenarios. First, we obtain a general capacity inner bound including various previous results as its special cases, where, the informed encoder transmits two-layer individual description of side information to the relay and destination, and block Markov, superposition, Marton and Gel’fand-Pinsker coding at the encoders and sliding-window decoding at the receiver are exploited. This individual side information can be considered as the partial SI at the relay and destination, which can potentially increase the transmission rates, especially for the uninformed encoder. Second, to show practical importance of the obtained inner bound, we extend it to the Gaussian version, which can be used in studying the coverage region and energy efficiency. Third, we study outer bounding the capacity region, aiming at outer bounds less than cut set bounds, by considering a general MARC with partial decode and forward (PDF) strategy at the relay. Finally, the obtained Gaussian inner bound is evaluated numerically, and we showed that our capacity inner bound results in more sum-rate in comparison with the previous studies.

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Authors and Affiliations

  1. Department of Communication Engineering, University of Sistan and Baluchestan, Zahedan, Iran

    Jalil Etminan & Farahnaz Mohanna

  2. Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Ghosheh Abed Hodtani

Authors
  1. Jalil Etminan
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  2. Farahnaz Mohanna
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  3. Ghosheh Abed Hodtani
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Correspondence to Farahnaz Mohanna.

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Appendices

Appendices

In details for Appendix B, Appendix C and Appendix D:

  1. a

    : follows since Fano’s inequality.

  2. b

    : follows since Csiszar-Korners’s sum identity.

  3. c

    : follows since \(X_{Ri}-W_0W_1^{\prime }W_2^{\prime }S_{i+1}^nY_{D}^{i-1}-W_1^{\prime \prime } W_2^{\prime \prime }S_{i}\).

  4. d

    : follows since \(W_1^{\prime \prime }-W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }S_{i+1}^n-S_{i}\).

  5. e

    : follows since \(W_1^{\prime }-W_0W_2^{\prime }S_{i+1}^n-S_{i}\).

  6. f

    : follows since \(X_{Ri}=f\left( Y_{R}^{i-1}\right) \)

  7. g

    : follows since \(H\left( S_{i}\right) =H\left( S_i|S_{i+1}^n\right) \).

  8. h

    : follows since \(X_{1i}=f\left( W_0W_1^{\prime }W_1^{\prime \prime }\right) \).

  9. k

    : follows since \(W_1^{\prime \prime }-W_0W_1^{\prime }W_2^{\prime }Y_{D}^{i-1}S_{i+1}^n-S_{i}\).

  10. l

    : follows since the channel is degraded.

  11. n

    : follows since \(W_1^{\prime }W_1^{\prime \prime }-X_{1i}-Y_{Ri}Y_{Di}\).

  12. t

    : follows since \(X_{0i}=f\left( W_0\right) \), \(X_{1Di}=f\left( W_1\right) \) and \(X_{2Di}=f\left( W_2,S^n\right) \).

  13. q

    : follows since \(Y_{Ri}-X_{0i}S_iX_{Ri}-X_{1Di}X_{2Di}\) and \(Y_{Di}-X_{1Di}X_{2Di}S_iX_{Ri}-X_{0i}Y_{Ri}\).

  14. r

    : follows since \(Y_R^{i-1}X_{Ri}-Y_D^nW_0W_1W_2S^{i-1}-S_i\)

A Proof of the Theorem 2

By using the channel inputs distribution and auxiliary random variables defined in Sect. 3.2 we extended the results of the remark 1 to Gaussian continuous channel, Hence:

$$\begin{aligned}&h\left( Y_R|V_1V_2X_1Z_2{\hat{S}}_R\right) =h\left( {\tilde{Y}}_R|V_2Z_2{\hat{S}}_R\right) \\&\quad =\frac{1}{2}\log _2\left( 2\pi e E\left[ \left( {\tilde{Y}}_R-E\left( {\tilde{Y}}_R|V_2Z_2{\hat{S}}_R\right) \right) ^2\right] \right) \end{aligned}$$

Where, we define \({\tilde{Y}}_R=\frac{X_2}{d_{2R}^{\frac{\alpha }{2}}}+\frac{S}{d_{SR}^{\frac{\alpha }{2}}}+Z_R\) and \({\tilde{Y}}_D=\frac{X_2}{d_{2D}^{\frac{\alpha }{2}}}+\frac{V_{R_2}}{d_{RD}^{\frac{\alpha }{2}}}+\frac{S}{d_{SD}^{\frac{\alpha }{2}}}+Z_D\) and by using MMSE estimator it can be shown that:

$$\begin{aligned} E\left( {\tilde{Y}}_R|V_2Z_2{\hat{S}}_R\right)&=\frac{\frac{{\bar{\beta }}P_2\left( 1-\rho _{2R}^2\right) }{d_{2R}^{\frac{\alpha }{2}}}+\frac{\eta _2 D}{d_{SR}^{\frac{\alpha }{2}}}}{{\bar{\beta }}P_2\left( 1-\rho _{2R}^2\right) +\eta _2^2D}Z_2\\&\quad +\sqrt{\frac{{\bar{\beta }}P_2}{{\bar{\gamma }}P_R d_{2R}^{\alpha }}}V_2+\frac{{\hat{S}}_R}{d_{SR}^{\frac{\alpha }{2}}} \end{aligned}$$

And consequently, it can be shown that:

$$\begin{aligned}&h\left( Y_R|V_1V_2X_1Z_2{\hat{S}}_R\right) =\frac{1}{2}\log _2\left( 2 \pi e r_3\right) \\&h\left( Y_R|V_1V_2Z_2{\hat{S}}_R\right) =\frac{1}{2}\log _2\left( 2 \pi e \left( \frac{P_1\left( 1-\rho _{1R}^2\right) }{d_{1R}^{\alpha }}+r_3\right) \right) \\&h\left( Y_R|V_1V_2X_1{\hat{S}}_R\right) =\frac{1}{2}\log _2\left( 2 \pi e r_1\right) \\&h\left( Y_R|V_1V_2 {\hat{S}}_R\right) =\frac{1}{2}\log _2\left( 2 \pi e r_2\right) \\&h\left( Y_R|V_1V_2X_1Z_2Q_R {\hat{S}}_R\right) =\frac{1}{2}\log _2\left( 2 \pi e r_4\right) \end{aligned}$$

Also, similarly it can be shown that:

$$\begin{aligned}&h\left( Y_D\right) =\frac{1}{2}\log _2\left( 2 \pi e D_1\right) ,h\left( Y_D|V_1\right) =\frac{1}{2}\log _2\left( 2 \pi e D_2\right) \\&h\left( Y_D|V_1V_2\right) =\frac{1}{2}\log _2\left( 2 \pi e D_3\right) \\&h\left( Y_D|V_1V_2X_1\right) =\frac{1}{2}\log _2\left( 2 \pi e D_4\right) \\&h\left( Y_D|V_1V_2X_1Z_2\right) =\frac{1}{2}\log _2\left( 2 \pi e D_5\right) \\&h\left( Y_D|V_1V_2Z_2\right) =\frac{1}{2}\log _2\left( 2 \pi e \left( \frac{P_1 \left( 1-\rho _{1R}^2\right) }{d_{1D}^{\alpha }}+D_5\right) \right) \\&h(Q_R|V_2Z_2{\hat{S}}_R)=\frac{1}{2}\log _2 (2 \pi e (\beta P_2+\frac{\eta _3^2 D {\bar{\beta }}P_2(1-\rho _{2R}^2)}{{\bar{\beta }}P_2(1-\rho _{2R}^2)+\eta _2^2D}))\\&h\left( Q_R|V_2Z_2{\hat{S}}_RS\right) =\frac{1}{2}\log _2\left( 2 \pi e \beta P_2\right) \\&h\left( Z_2|V_2{\hat{S}}_R\right) =\frac{1}{2}\log _2\left( 2 \pi e\left( {\bar{\beta }}P_2 \left( 1-\rho _{2R}^2\right) +\eta _2^2 D\right) \right) \\&h\left( Z_2|V_2 {\hat{S}}_RS\right) =\frac{1}{2}\log _2 \left( 2 \pi e \left( {\bar{\beta }}P_2\left( 1-\rho _{2R}^2\right) \right) \right) \\&h\left( V_2\right) =\frac{1}{2}\log _2 \left( 2 \pi e\left( {\bar{\gamma }}P_R+\eta _1^2\left( Q-D\right) \right) \right) \\&h\left( V_2|{\hat{S}}_R\right) =\frac{1}{2}\log _2\left( 2 \pi e {\bar{\gamma }}P_R\right) \end{aligned}$$

And the proof is completed.

B Proof of Theorem 3

$$\begin{aligned}&nR_0=H\left( W_0\right) \le H\left( W_0W_1^{\prime }W_2^{\prime }\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} I\left( W_0W_1^{\prime }W_2^{\prime };Y_{R}^nY_D^n\right) -I\left( W_0W_1^{\prime }W_2^{\prime };S^n\right) +n\epsilon _n \nonumber \\&\quad \cong \sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime };Y_{Ri}Y_{Di}|Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{f}}\sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_{R}^{i-1}Y_{D}^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. - I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{b}}\sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_{R}^{i-1}Y_{D}^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. - I\left( Y_{R}^{i-1}Y_{D}^{i-1};S_{i}|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{f,g}}\sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_{R}^{i-1}Y_{D}^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. - I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^nY_{R}^{i-1}Y_{D}^{i-1}X_{Ri};S_{i}\right) \right] \nonumber \\&\quad \le \sum _{i=1}^n{\left[ I\left( U_{0i}U_{1i}U_{2i};Y_{Ri}Y_{Di}|Q_{i}X_{Ri}\right) -I\left( U_{2i};S_i|U_{0i}X_{Ri}\right) \right] } \nonumber \\&nR_0=H\left( W_0\right) \le H\left( W_0W_1^{\prime }W_2^{\prime }\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} I\left( W_0W_1^{\prime }W_2^{\prime };Y_D^n\right) -I\left( W_0W_1^{\prime }W_2^{\prime };S^n\right) +n\epsilon _n \nonumber \\&\quad \cong \sum _{i=1}^n{\left[ I\left( W_0W_1^{\prime }W_2^{\prime };Y_{Di}|Y_{D}^{i-1}\right) -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] } \nonumber \\&\quad =\sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Di}|Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. - I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_{D}^{i-1}\right) -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \end{aligned}$$
(95)
$$\begin{aligned}&\quad {\mathop {=}\limits ^{b}}\sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Di}|Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. - I\left( Y_{D}^{i-1};S_{i}|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c,g}}\sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime }Y_{D}^{i-1}X_{Ri}S_{i+1}^n;Y_{Di}\right) \right. \nonumber \\&\qquad \left. - I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^nY_{D}^{i-1}X_{Ri};S_{i}\right) \right] \nonumber \\&\quad \le \sum _{i=1}^n\left[ I\left( W_0W_1^{\prime }W_2^{\prime }Y_{D}^{i-1}X_{Ri}S_{i+1}^n;Y_{Di}\right) \right. \nonumber \\&\qquad \left. - I\left( W_2^{\prime }S_{i+1}^n;S_{i}|W_0Y_{D}^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad =\sum _{i=1}^n{\left[ I\left( U_{0i}U_{1i}U_{2i}X_{Ri};Y_{Di}\right) -I\left( U_{2i};S_i|U_{0i}X_{Ri}\right) \right] } \nonumber \\&nR_{1}=H\left( W_1^{\prime }|W_0W_2^{\prime }\right) +H\left( W_1^{\prime \prime }|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}}I\left( W_1^{\prime };Y_R^nY_D^n|W_0W_2^{\prime }\right) +I\left( W_1^{\prime \prime };Y_D^n|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }\right) \nonumber \\&\qquad +n\epsilon _n\cong \sum _{i=1}^n \left[ I\left( W_1^{\prime };Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }Y_{D}^{i-1}\right) \right] \nonumber \\&\quad = \sum _{i=1}^n \left[ I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime };Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }W_1^{\prime \prime }Y_{D}^{i-1}\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{b}} \sum _{i=1}^n \left[ I\left( Y_{R}^{i-1}Y_{D}^{i-1};S_i|W_0W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime };Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{R}^{i-1}Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }W_1^{\prime \prime }S_{i+1}^n\right) \right] \nonumber \\&\quad \le \sum _{i=1}^n \left[ I\left( W_1^{\prime }Y_{R}^{i-1}Y_{D}^{i-1};S_i|W_0W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime };Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime }Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{R}^{i-1}Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }W_1^{\prime \prime }S_{i+1}^n\right) \right] \end{aligned}$$
(96)
$$\begin{aligned}&\quad {\mathop {\le }\limits ^{d,e,f}} \sum _{i=1}^n \left[ I\left( W_1^{\prime };Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime };X_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c}} \sum _{i=1}^n \left[ I\left( W_1^{\prime };Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( X_{1i}W_1^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }X_{Ri}Y_{D}^{i-1}S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{n}} \sum _{i=1}^n \left[ I\left( W_1^{\prime };Y_{Ri}Y_{Di}|W_0W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( X_{1i};Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_2^{\prime \prime }X_{Ri}Y_{D}^{i-1}S_{i+1}^n\right) \right] \nonumber \\&\quad = \sum _{i=1}^n \left[ I\left( U_{1i};Y_{Ri}Y_{Di}|U_{0i}U_{2i}X_{Ri}Q_{i}\right) \right. \nonumber \\&\qquad \left. +I\left( X_{1i};Y_{Di}|U_{0i}U_{1i}U_{2i}X_{Ri}V_{i}\right) \right] \nonumber \\&nR_{2}=H\left( W_2^{\prime }|W_0W_1^{\prime }\right) +H\left( W_2^{\prime \prime }|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}}I\left( W_2^{\prime };Y_R^nY_D^n|W_0W_1^{\prime }\right) +I\left( W_2^{\prime \prime };Y_D^n|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }\right) \nonumber \\&\qquad -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime };S^n\right) +n\epsilon _n \nonumber \\&\quad \cong \sum _{i=1}^n \left[ I\left( W_2^{\prime };Y_{Ri}Y_{Di}|W_0W_1^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad =\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime }S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{b}}\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{R}^{i-1}Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{f,g}}\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime }Y_{D}^{i-1};S_{i}|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime }S_{i+1}^nY_{D}^{i-1};S_i\right) \right] \end{aligned}$$
(97)
$$\begin{aligned}&\quad {\mathop {\le }\limits ^{c,k,h}}\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime };X_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }X_{1i}Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime }S_{i+1}^nY_{D}^{i-1}X_{Ri};S_i\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c}}\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }Y_{R}^{i-1}Y_{D}^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime }W_2^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }X_{1i}Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_2^{\prime }W_2^{\prime \prime }S_{i+1}^n;S_i|W_0Y_{D}^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{n}}\sum _{i=1}^n\left[ I\left( U_{2i};Y_{Ri}Y_{Di}|U_{0i}U_{1i}Q_iX_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( V_i;Y_{Di}|U_{0i}U_{1i}U_{2i}X_{1i}X_{Ri}\right) -I\left( U_{2i}V_{i};S_i|U_{0i}X_{Ri}\right) \right] \end{aligned}$$
(98)
$$\begin{aligned}&nR_{2}=H\left( W_2^{\prime }|W_0W_1^{\prime }\right) +H\left( W_2^{\prime \prime }|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} \sum _{i=1}^n \left[ I\left( W_2^{\prime };Y_{Di}|W_0W_1^{\prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad =\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime }S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{b,g}}\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^nX_{Ri};Y_{Di}|W_0W_1^{\prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime }Y_{D}^{i-1};S_{i}|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime }S_{i+1}^nY_{D}^{i-1};S_i\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c,k,h}}\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^nX_{Ri};Y_{Di}|W_0W_1^{\prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2^{\prime \prime };X_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }X_{1i}Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_1^{\prime \prime }W_2^{\prime \prime }S_{i+1}^nY_{D}^{i-1}X_{Ri};S_i\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c}}\sum _{i=1}^n\left[ I\left( W_2^{\prime }S_{i+1}^nX_{Ri};Y_{Di}|W_0W_1^{\prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime }W_2^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }X_{1i}Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_2^{\prime }W_2^{\prime \prime }S_{i+1}^n;S_i|W_0Y_{D}^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{n}}\sum _{i=1}^n\left[ I\left( U_{2i}X_{Ri};Y_{Di}|U_{0i}U_{1i}\right) -I\left( U_{2i}V_{i};S_i|U_{0i}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( V_i;Y_{Di}|U_{0i}U_{1i}U_{2i}X_{1i}X_{Ri}\right) \right] \end{aligned}$$
(99)
$$\begin{aligned}&n\left( R_0+R_1\right) =H\left( W_0W_1^{\prime }\right) +H\left( W_{1}^{\prime \prime }|W_0W_1^{\prime }\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} \sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime };Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_{1}^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}\right) -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad = \sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_{1}^{\prime \prime }S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. - I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_{1}^{\prime \prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{b,f}}\sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_{1}^{\prime \prime };X_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. - I\left( Y_R^{i-1}Y_D^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_{i}|W_0W_1^{\prime }W_{1}^{\prime \prime }W_2^{\prime }W_{2}^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c}}\sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1^{\prime \prime }Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_{1}^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_{i}|W_0W_1^{\prime }W_{1}^{\prime \prime }W_2^{\prime }W_{2}^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }Y_R^{i-1}Y_D^{i-1};S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c,g,f}}\sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( X_{1i}W_{1}^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }Y_R^{i-1}X_{Ri}Y_D^{i-1}S_{i+1}^n;S_i\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{n}}\sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( X_{1i};Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_2^{\prime }S_{i+1}^n;S_i|W_0Y_D^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad = \sum _{i=1}^n \left[ I\left( U_{0i}U_{1i}U_{2i};Y_{Ri}Y_{Di}|Q_iX_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( X_{1i};Y_{Di}|U_{0i}U_{1i}U_{2i}X_{Ri}V_i\right) -I\left( U_{2i};S_i|U_{0i}X_{Ri}\right) \right] \nonumber \\&n\left( R_0+R_2\right) =H\left( W_0W_2^{\prime }\right) +H\left( W_{2}^{\prime \prime }|W_0W_2^{\prime }\right) \nonumber \\&\quad \le H\left( W_0W_1^{\prime }W_2^{\prime }\right) +H\left( W_{2}^{\prime \prime }|W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} \sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime };Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_{2}^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }W_{2}^{\prime \prime };S_i|S_{i+1}^n\right) \right] \end{aligned}$$
(100)
$$\begin{aligned}&\quad = \sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. +I\left( W_{2}^{\prime \prime }S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. - I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Di}|W_0W_1^{\prime }W_{1}^{\prime \prime }W_2^{\prime }W_{2}^{\prime \prime }Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }W_{2}^{\prime \prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{b,f}}\sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_{R}^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_{2}^{\prime \prime };X_{Ri}Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }Y_{D}^{i-1}S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. - I\left( Y_R^{i-1}Y_D^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{D}^{i-1};S_{i}|W_0W_1^{\prime }W_{1}^{\prime \prime }W_2^{\prime }W_{2}^{\prime \prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }W_{2}^{\prime \prime };S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{d,h}}\sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_{R}^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_{1}^{\prime \prime }Y_{D}^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. +I\left( W_{2}^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }Y_{D}^{i-1}S_{i+1}^nX_{Ri}\right) \right. \nonumber \\&\qquad \left. - I\left( Y_D^{i-1};S_i|W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1^{\prime }W_2^{\prime }W_{1}^{\prime \prime }W_{2}^{\prime \prime }Y_{D}^{i-1};S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c,g,k,h}}\sum _{i=1}^{n}\left[ I\left( W_0W_1^{\prime }W_2^{\prime }S_{i+1}^n;Y_{Ri}Y_{Di}|Y_{R}^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_{1}^{\prime \prime }W_{2}^{\prime \prime };Y_{Di}|W_0W_1^{\prime }W_2^{\prime }Y_{D}^{i-1}S_{i+1}^nX_{Ri}X_{1i}\right) \right. \nonumber \\&\qquad \left. -I\left( W_2^{\prime }W_{2}^{\prime \prime }S_{i+1}^n;S_i|W_0Y_{D}^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{n}}\sum _{i=1}^n\left[ I\left( U_{0i}U_{1i}U_{2i};Y_{Ri}Y_{Di}|Q_iX_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( V_i;Y_{Di}|U_{0i}U_{1i}U_{2i}X_{Ri}X_{1i}\right) -I\left( U_{2i}V_i;S_i|U_{0i}X_{Ri}\right) \right] \end{aligned}$$
(101)

For brevity, only some bounds are proved. The other bounds can be done similarly.

C Proof of Theorem 4

$$\begin{aligned}&nR_2=H\left( W_2\right) =H\left( W_2|W_0W_1\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} \sum _{i=1}^n \left[ I\left( W_2;Y_{Ri}Y_{Di}|W_0W_1Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad =\sum _{i=1}^n\left[ I\left( W_2S_{i+1}^nS_i;Y_{Ri}Y_{Di}|W_0W_1Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1W_2Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_{Ri}Y_{Di}|W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{b,f}}\sum _{i=1}^n \left[ I\left( W_2S_{i+1}^nS_i;Y_{Ri}Y_{Di}|W_0W_1Y_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I(Y_R^{i-1}Y_D^{i-1}X_{Ri};S_i|W_0W_1W_2S_{i+1}^n)-I(W_0W_1W_2;S_i|S_{i+1}^n) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_{Ri}Y_{Di}|W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{g}}\sum _{i=1}^n \left[ I\left( W_2S_{i+1}^nS_i;Y_{Ri}|W_0W_1Y_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_2S_{i+1}^nS_i;Y_{Di}|W_0W_1Y_R^{i-1}Y_D^{i-1}Y_{Ri}X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}X_{Ri}Y_{Ri}Y_{Di};S_i\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{l}}\sum _{i=1}^n \left[ I\left( W_2S_{i+1}^nS_iY_R^{i-1};Y_{Ri}|W_0W_1Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}X_{Ri}Y_{Ri}Y_{Di};S_i\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{c,h}}\sum _{i=1}^n \left[ I\left( W_2S_{i+1}^n;Y_{Ri}|W_0W_1Y_D^{i-1}X_{Ri}X_{1i}\right) \right. \nonumber \\&\qquad \left. +I\left( S_i;Y_{Ri}|W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( Y_R^{i-1};Y_{Ri}|W_0W_1W_2S_{i+1}^nS_iY_D^{i-1}X_{Ri}X_{1i}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri};S_i\right) \right. \nonumber \\&\qquad \left. -I\left( Y_{Ri};S_i|W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( Y_R^{i-1}Y_{Di};S_i|W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri}Y_{Ri}\right) \right] \nonumber \\&\quad \le \sum _{i=1}^n \left[ I\left( W_1W_2S_{i+1}^n;Y_{Ri}|W_0Y_D^{i-1}X_{Ri}X_{1i}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1Y_R^{i-1};Y_{Ri}|W_0W_2S_{i+1}^nY_D^{i-1}X_{Ri}X_{1i}S_i\right) \right. \nonumber \\&\qquad \left. -I\left( W_2S_{i+1}^n;S_i|W_0Y_D^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{n}}\sum _{i=1}^n \left[ I\left( U_{2i};Y_{Ri}|U_{0i}X_{1i}X_{Ri}\right) -I\left( U_{2i};S_i|U_{0i}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( Q_i;Y_{Ri}|U_{0i}X_{1i}U_{2i}X_{Ri}S_i\right) \right] \end{aligned}$$
(102)
$$\begin{aligned}&n\left( R_1+R_2\right) =H\left( W_1W_2\right) =H\left( W_1W_2|W_0\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} \sum _{i=1}^n \left[ I\left( W_1W_2;Y_{Ri}Y_{Di}|W_0Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad =\sum _{i=1}^n\left[ I\left( W_1W_2S_{i+1}^nS_i;Y_{Ri}Y_{Di}|W_0Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1W_2Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_{Ri}Y_{Di}|W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{b,f}}\sum _{i=1}^n \left[ I\left( W_1W_2S_{i+1}^nS_i;Y_{Ri}Y_{Di}|W_0Y_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( Y_R^{i-1}Y_D^{i-1}X_{Ri};S_i|W_0W_1W_2S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_{Ri}Y_{Di}|W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&{\mathop {\le }\limits ^{l,g}}\sum _{i=1}^n \left[ I\left( W_1W_2S_{i+1}^nS_iY_R^{i-1};Y_{Ri}|W_0Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \qquad \left. -I\left( W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}X_{Ri}Y_{Ri}Y_{Di};S_i\right) \right] \nonumber \\&{\mathop {\le }\limits ^{c,h}}\sum _{i=1}^n \left[ I\left( X_{1i}W_1W_2S_{i+1}^n;Y_{Ri}|W_0Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \qquad \left. +I\left( Y_R^{i-1};Y_{Ri}|W_0W_1W_2S_{i+1}^nS_iY_D^{i-1}X_{Ri}X_{1i}\right) \right. \nonumber \\&\qquad \qquad \left. -I\left( W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri};S_i\right) \right. \nonumber \\&\qquad \qquad \left. -I\left( Y_R^{i-1}Y_{Di};S_i|W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri}Y_{Ri}\right) \right] \nonumber \\&\le \sum _{i=1}^n \left[ I\left( X_{1i}W_1W_2S_{i+1}^n;Y_{Ri}|W_0Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \qquad \left. +I\left( W_1Y_R^{i-1};Y_{Ri}|W_0W_2S_{i+1}^nY_D^{i-1}X_{Ri}X_{1i}S_i\right) \right. \nonumber \\&\qquad \qquad \left. -I\left( W_2S_{i+1}^n;S_i|W_0Y_D^{i-1}X_{Ri}\right) \right] \nonumber \\&{\mathop {=}\limits ^{n}}\sum _{i=1}^n \left[ I\left( X_{1i}U_{2i};Y_{Ri}|U_{0i}X_{Ri}\right) -I\left( U_{2i};S_i|U_{0i}X_{Ri}\right) \right. \nonumber \\&\qquad \qquad \left. +I\left( Q_i;Y_{Ri}|U_{0i}X_{1i}U_{2i}X_{Ri}S_i\right) \right] \end{aligned}$$
(103)
$$\begin{aligned}&n\left( R_0+R_1+R_2\right) =H\left( W_0W_1W_2\right) \nonumber \\&\quad {\mathop {\le }\limits ^{a}} \sum _{i=1}^n \left[ I\left( W_0W_1W_2;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad =\sum _{i=1}^n\left[ I\left( W_0W_1W_2S_{i+1}^nS_i;Y_{Ri}Y_{Di}|Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_{i+1}^n;Y_{Ri}Y_{Di}|W_0W_1W_2Y_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_{Ri}Y_{Di}|W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{b,f}}\sum _{i=1}^n \left[ I\left( W_0W_1W_2S_{i+1}^nS_iY_R^{i-1}Y_D^{i-1};Y_{Ri}Y_{Di}|X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( Y_R^{i-1}Y_D^{i-1}X_{Ri};S_i|W_0W_1W_2S_{i+1}^n\right) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_{Ri}Y_{Di}|W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2;S_i|S_{i+1}^n\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{g,l}}\sum _{i=1}^n \left[ I\left( W_0W_1W_2S_{i+1}^nS_iY_R^{i-1}Y_D^{i-1};Y_{Ri}|X_{Ri}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2S_{i+1}^nY_R^{i-1}Y_D^{i-1}X_{Ri}Y_{Ri}Y_{Di};S_i\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{c,h}}\sum _{i=1}^n \left[ I\left( X_{1i}W_0W_1W_2S_{i+1}^nY_D^{i-1};Y_{Ri}|X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( Y_R^{i-1};Y_{Ri}|W_0W_1W_2X_{1i}S_{i+1}^nS_iY_D^{i-1}X_{Ri}X_{1i}\right) \right. \nonumber \\&\qquad \left. -I\left( W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri};S_i\right) \right. \nonumber \\&\qquad \left. -I\left( Y_R^{i-1}Y_{Di};S_i|W_0W_1W_2S_{i+1}^nY_D^{i-1}X_{Ri}Y_{Ri}\right) \right] \nonumber \\&\quad \le \sum _{i=1}^n \left[ I\left( X_{1i}W_0W_1W_2S_{i+1}^nY_D^{i-1};Y_{Ri}|X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( W_1Y_R^{i-1};Y_{Ri}|W_0W_2S_{i+1}^nY_D^{i-1}X_{Ri}X_{1i}S_i\right) \right. \nonumber \\&\qquad \left. -I\left( W_2S_{i+1}^n;S_i|W_0Y_D^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad {\mathop {=}\limits ^{n}}\sum _{i=1}^n \left[ I\left( X_{1i}U_{2i}U_{0i};Y_{Ri}|X_{Ri}\right) -I\left( U_{2i};S_i|U_{0i}X_{Ri}\right) \right. \nonumber \\&\qquad \left. +I\left( Q_i;Y_{Ri}|U_{0i}X_{1i}U_{2i}X_{Ri}S_i\right) \right] \end{aligned}$$
(104)

For brevity, only some bounds are proved. The other bounds can be done similarly.

D Proof of Theorem 5

$$\begin{aligned}&n\left( R_1+R_2\right) =H\left( W_1W_2\right) =H\left( W_1W_2|W_0\right) \nonumber \\&\quad \le I\left( W_1W_2;Y_R^nY_D^nS^n|W_0\right) +n\epsilon _n \nonumber \\&\quad \le \sum _{i=1}^n \left[ I\left( W_1W_2;Y_{Ri}Y_{Di}|W_0S^nY_{R}^{i-1}Y_D^{i-1}\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{f,t,q}}\sum _{i=1}^n \left[ I\left( X_{1Di}X_{2Di};Y_{Di}|S_iX_{0i}X_{Ri}\right) \right] \end{aligned}$$
(105)
$$\begin{aligned}&n\left( R_0+R_1+R_2\right) =H\left( W_0W_1W_2\right) \nonumber \\&\quad \le \sum _{i=1}^n \left[ I\left( W_0W_1W_2;Y_{Ri}Y_{Di}|S^nY_{R}^{i-1}Y_D^{i-1}\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{f,t,q}}\sum _{i=1}^n \left[ I\left( X_{0i}X_{1Di}X_{2Di};Y_{Ri}Y_{Di}|S_iX_{Ri}\right) \right] \nonumber \\&\quad =\sum _{i=1}^n \left[ I\left( X_{0i}X_{1Di}X_{2Di};Y_{Ri}|S_iX_{Ri}\right) \right. \nonumber \\&\qquad \left. + I\left( X_{0i}X_{1Di}X_{2Di};Y_{Di}|S_iX_{Ri}Y_{Ri}\right) \right] \\&\quad {\mathop {\le }\limits ^{q}}\sum _{i=1}^n \left[ I\left( X_{0i};Y_{Ri}|S_i X_{Ri}\right) +I\left( X_{1Di}X_{2Di};Y_{Di}|S_iX_{Ri}\right) \right] \nonumber \end{aligned}$$
(106)
$$\begin{aligned}&n\left( R_0+R_1+R_2\right) =H\left( W_0W_1W_2\right) \le I\left( W_0W_1W_2;Y_D^n\right) \nonumber \\&\qquad +n\epsilon _n=I\left( W_0W_1W_2S^n;Y_D^n\right) -I\left( S^n;Y_D^n|W_0W_1W_2\right) \nonumber \\&\quad =\sum _{i=1}^n \left[ I\left( W_0W_1W_2S^n;Y_{Di}|Y_{D}^{i-1}\right) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_D^n|W_0W_1W_2S^{i-1}\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{r}}\sum _{i=1}^n \left[ I\left( W_0W_1W_2S^nY_{D}^{i-1}X_{1Di}X_{2Di}X_{Ri};Y_{Di}\right) \right. \nonumber \\&\qquad \left. -I\left( S_i;Y_D^nW_0W_1W_2S^{i-1}Y_{R}^{i-1}X_{Ri}\right) \right] \nonumber \\&\quad {\mathop {\le }\limits ^{q}}\sum _{i=1}^n \left[ I\left( X_{Ri};Y_{Di}\right) +I\left( X_{1Di}X_{2Di};Y_{Di}|X_{Ri}S_i\right) \right] \end{aligned}$$
(107)

For brevity, only some bounds are proved. The other bounds can be done similarly.

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Etminan, J., Mohanna, F. & Abed Hodtani, G. Inner and outer bounding capacity region for multiple-access relay channel with non-causal side information at one encoder. Wireless Netw 28, 85–106 (2022). https://doi.org/10.1007/s11276-021-02815-8

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