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Matrix differentiation for capacity region of Gaussian multiple access channels under weighted total power constraint

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Abstract

Maximization the capacity region of Gaussian multiple access channels with vector inputs and vector outputs has been extensively studied in existing schemes. Although these schemes are proven effective in various real-life applications, they are inapplicable to deal with channels with matrix variables subjected to certain constraints. In this work, we present a new framework to estimate the capacity region of Gaussian multiple access channels with matrix inputs and outputs under weighted total power constraints. We propose an optimization model to address this issue and prove its concavity. By introducing an I-chain rule for matrix differentiation, the gradient of the objective function involving matrix variables can be obtained. An algorithm, named normalized projected gradient method (NPGM) is developed to find the global optimal solution for the proposed model. The convergence of NPGM is established by utilizing projection and normalization operators. Simulation results provide an interesting insight that NPGM can manage the existent situations within an unified framework, and provide a novel universal technical solution to optimize the capacity region under weighted total power constraints.

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References

  1. Abadir KM, Magnus JR (2005) Matrix algebra, vol. 1. Cambridge University Press

  2. Ahlswede R (1974) The capacity region of a channel with two senders and two receivers. Ann Probab 29 (5):805–814

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnes RJ (2006) Matrix differentiation. Springs Journal

  4. Basher U, Shirazi A, Permuter HH (2012) Capacity region of finite state multiple-access channels with delayed state information at the transmitters. IEEE Trans Inf Theory 58(6):3430–3452

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng RS, Verdú S (1993) Gaussian multiaccess channels with ISI: capacity region and multiuser water-filling. IEEE Trans Inf Theory 39(3):773–785

    Article  MathSciNet  MATH  Google Scholar 

  6. Cover TM, Leung CSK (1981) An achievable rate region for the multiple-access channel with feedback. IEEE Trans Inf Theory 27(3):292–298

    Article  MathSciNet  MATH  Google Scholar 

  7. Harshan J, Rajan BS (2008) Finite signal-set capacity of two-user Gaussian multiple access channel. In: IEEE International symposium on information theory, pp 1203–1207

  8. den Hertog D, Roos K (2009) Computing safe dike heights at minimal costs. Tech. rep., centER Applied Research Report Tilburg University

  9. Huang Q, Yang H (2013) Optimal power control for weighted sum rate of multiple access channel. Int J Digit Cont Tec App 7(5):247–254

    Google Scholar 

  10. Hui JYH, Humblet PA (1985) The capacity region of the totally asynchronous multiple-access channel. IEEE Trans Inf Theory 31(2):207–216

    Article  MATH  Google Scholar 

  11. Jindal N, Vishwanath S, Goldsmith A (2004) On the duality of gaussian multiple-access and broadcast channels. IEEE Trans Inf Theory 50(5):768–783

    Article  MathSciNet  MATH  Google Scholar 

  12. Li G, Ding J, Wen C, Pei J (2016) Optimal control of complex networks based on matrix differentiation. EPL Europhysics Letters 115(6):68,005

    Article  Google Scholar 

  13. Li G, Hu W, Xiao G, Deng L, Tang P, Pei J, Shi L (2016) Minimum-cost control of complex networks. New J Phys 18(1): 1–3

    Google Scholar 

  14. Liu R, Poor HV (2009) Secrecy capacity region of a multiple-antenna Gaussian broadcast channel with confidential messages. IEEE Trans Inf Theory 55(3):1235–1249

    Article  MathSciNet  MATH  Google Scholar 

  15. Magnus JR, Neudecker H (1995) Matrix differential calculus with applications in statistics and econometrics

  16. Petersen KB, Pedersen MS (2012) The matrix cookbook. Tech rep

  17. Sankar L, Mandayam NB, Poor HV (2009) On the sum-capacity of degraded Gaussian multiple-access relay channels. IEEE Trans Inf Theory 55(12):5394–5411

    Article  MathSciNet  MATH  Google Scholar 

  18. Shi Y, Hou YT (2008) On the capacity of UWB-based wireless sensor networks. Comput Netw 52:2797–2804

    Article  MATH  Google Scholar 

  19. Verdú S (1986) Minimum probability of error for asynchronous Gaussian multiple-access channels. IEEE Trans Inf Theory 32(1):85–96

    Article  MathSciNet  MATH  Google Scholar 

  20. Verdú S (1989) The capacity region of the symbol-asynchronous Gaussian multiple-access channel. IEEE Trans Inf Theory 35(4):733–751

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang T, Seyedi A, Heinzelman AVW (2013) Optimal rate allocation for distributed source coding over Gaussian multiple access channels. IEEE Trans Wireless Commun 12(5):2002–2013

    Article  Google Scholar 

  22. Werner-Allen G, Lorincz K, Welsh M, Marcillo O, Ruiz JJM, Lees J (2006) Deploying a wireless sensor network on an active volcano. IEEE Internet Computing 10(2):18–25

    Article  Google Scholar 

  23. Wimmer HK (1988) External problems for Hölder norms of matrices and realizations of linear systems. Siam J Matrix Anal Appl 9(3):314–322

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu R, Wang H, Chen M, Tian C (2014) Matrix division multiple access for mini centralized network. In: IEEE International conference on communications 2014-wireless communications symposium, 5914–5919

  25. Yu W, Rhee W, Boyd S, Cioffi JM (2004) Iterative water-filling for Gaussian vector multiple-access channels. IEEE Trans Inf Theory 50(1):145–152

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Key Laboratory of Environment and Control for Flight Vehicle, School of Aerospace, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Zhao-Xu Yang & Hai-Jun Rong

  2. School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Guang-She Zhao

  3. Center for Brain Inspired Computing Research, Department of Precision Instrument, Tsinghua University, Beijing, 100084, People’s Republic of China

    Guoqi Li

Authors
  1. Zhao-Xu Yang
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  2. Guang-She Zhao
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  4. Hai-Jun Rong
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Corresponding author

Correspondence to Guoqi Li.

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Funding Information

This work is funded in part by the National Basic Research Program of China (Grant No. 2015CB057406), Beijing Natural Science Foundation (4164086), National Natural Science Foundation of China (61603209), and Independent Research Plan of Tsinghua University (20151080467).

Appendix A: Proof of Lemma 7

Appendix A: Proof of Lemma 7

Proof

Denote X j, l k as the lk-th element of matrix X j ,we use I-chain rule to calculate the derivative of \(\frac {\partial F(X)}{\partial X_{j,lk}}\).According to Lemma 5, we have

$$\begin{array}{@{}rcl@{}} \frac{\partial F(X)}{\partial X_{j,lk}} &=&\frac{\partial\left( \frac{1}{2}\log\left|\mathcal{G}\right|-\frac{1}{2}\log|Z|\right)}{\partial X_{j,lk}} =\frac{1}{2}\frac{\partial\left( \log\left|\mathcal{G}\right|\right)}{\partial X_{j,lk}}\\ &=&\frac{1}{2}\sum\limits^{\mathrm{m}}_{m=1}\sum\limits^{\mathrm{m}}_{n=1}\frac{\partial\left( \log\left|\mathcal{G}\right|\right)} {\partial\left( \mathcal{G}\right)_{mn} }\frac{\partial\left( \mathcal{G}\right)_{mn}}{\partial X_{j,lk}}. \end{array} $$
(52)

Then, we need to drive \(\frac {\partial \left (\log \left |\mathcal {G}\right |\right )} {\partial \left (\mathcal {G}\right )_{mn} }\)and \(\frac {\partial \left (\mathcal {G}\right )_{mn}}{\partial X_{j,lk}}\).

Using Lemma 5, it is obtained that

$$ \frac{\partial\left( \log\left|\mathcal{G}\right|\right)} {\partial\left( \mathcal{G}\right)_{mn} } =\sum\limits^{\mathrm{m}}_{f=1}\sum\limits^{\mathrm{m}}_{z=1} \frac{\partial\left( \log\left|\mathcal{G}\right|\right)}{\partial\left( \mathcal{G}\right)_{zf}} \frac{\partial\left( \mathcal{G}\right)_{zf}}{\partial\left( \mathcal{G}\right)_{mn}}. $$
(53)

For matrix A, a special case of differentiation (see [16]) is

$$ \frac{\partial \log\left|A\right|}{\partial A}=(A^{-1})^{T}. $$
(54)

Refer to Eq. 54, the left fraction term of Eq. 53becomes

$$ \frac{\partial\left( \log\left|\mathcal{G}\right|\right)}{\partial\left( \mathcal{G}\right)_{zf}} =\left( \left( \mathcal{G}\right)^{-1}\right)^{T}_{zf} =\left( \left( \mathcal{G}\right)^{-1}\right)_{fz}. $$
(55)

Using Lemma 6, the right fraction term of Eq. 53becomes

$$ \frac{\partial\left( \mathcal{G}\right)_{zf}}{\partial\left( \mathcal{G}\right)_{mn}} =\delta_{mz}\delta_{fn}. $$
(56)

By substituting Eqs. 55and 56into Eq. 53, we have

$$\begin{array}{@{}rcl@{}} \frac{\partial\left( \log\left|\mathcal{G}\right|\right)} {\partial\left( \mathcal{G}\right)_{mn} } &=&\sum\limits^{\mathrm{m}}_{f=1}\sum\limits^{\mathrm{m}}_{z=1} \left( \left( \mathcal{G}\right)^{-1}\right)_{fz} \delta_{mz}\delta_{fn} \\ &=&\sum\limits^{\mathrm{m}}_{f=1} \left( \left( \mathcal{G}\right)^{-1}\right)_{fm} \delta_{fn} = \left( \left( \mathcal{G}\right)^{-1}\right)_{nm}. \end{array} $$
(57)

Likewise, using Lemma 5, also we can obtain that

$$\begin{array}{@{}rcl@{}} &&\frac{\partial\left( \mathcal{G}\right)_{mn}}{\partial X_{j,lk}}\\ &=&\sum\limits^{\mathrm{m}}_{v=1}\sum\limits^{\mathrm{m}}_{c=1}\frac{\partial\left( \sum\limits^{K}_{i=1}H_{i}{X_{i}^{T}}X_{i}{H_{i}^{T}}+Z\right)_{mn}}{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}+Z\right)_{cv}} \\ &&\qquad\quad\frac{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}+Z\right)_{cv}}{\partial X_{j,lk}}\\ &=&\sum\limits^{\mathrm{m}}_{v=1}\sum\limits^{\mathrm{m}}_{c=1}\frac{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}+Z\right)_{mn}} {\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}+Z\right)_{cv}}\\ &&\qquad\quad\frac{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}+Z\right)_{cv}}{\partial X_{j,lk}}. \end{array} $$
(58)

Using Lemma 6, the left fraction term of Eq. 58becomes

$$ \frac{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}+Z\right)_{mn}} {\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}+Z\right)_{cv}}=\delta_{cm}\delta_{nv}. $$
(59)

By substituting Eq. 59into Eq. 58, we have

$$\begin{array}{@{}rcl@{}} &&\frac{\partial\left( \sum\limits^{K}_{i=1}H_{i}{X_{i}^{T}}X_{i}{H_{i}^{T}}+Z\right)_{mn}}{\partial X_{j,lk}}\\ &=&\sum\limits^{\mathrm{m}}_{v=1}\sum\limits^{\mathrm{m}}_{c=1} \left( \delta_{cm}\delta_{nv}\left( \frac{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}\right)_{cv}}{\partial X_{j,lk}}\right)\right). \end{array} $$
(60)

Using Lemma 5, the formula in parentheses of Eq. 60 becomes,

$$\begin{array}{@{}rcl@{}} \!\!&&\frac{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}\right)_{cv}}{\partial X_{j,lk}}\\ \!\!\!&=&\frac{\partial\sum\limits^{\mathrm{m}}_{a=1}(H_{j}{X_{j}^{T}})_{ca}(X_{j}{H_{j}^{T}})_{av}} {\partial X_{j,lk}}\\ \!\!\!&=&\sum\limits^{\mathrm{m}}_{a=1}\frac{\partial(H_{j}{X_{j}^{T}})_{ca}(X_{j}{H_{j}^{T}})_{av}} {\partial X_{j,lk}} \end{array} $$
(61)
$$\begin{array}{@{}rcl@{}} \!\!&=&\sum\limits^{\mathrm{m}}_{a=1}\frac{(H_{j}{X_{j}^{T}})_{ca}\partial(X_{j}{H_{j}^{T}})_{av}+(X_{j}{H_{j}^{T}})_{av}\partial(H_{j}{X_{j}^{T}})_{ca}} {\partial X_{j,lk}}\\ \!\!&=&\!\sum\limits^{\mathrm{m}}_{a=1}\!\left( \!(H_{j}{X_{j}^{T}})_{ca}\frac{\partial(X_{j}{H_{j}^{T}})_{av}} {\partial X_{j,lk}} \,+\,(X_{j}{H_{j}^{T}})_{av}\frac{\partial(H_{j}{\!X_{j}^{T}})_{ca}} {\partial X_{j,lk}} \right). \end{array} $$

Using Lemmas 5 and 6, the left fraction term in parentheses of Eq. 61 becomes,

$$\begin{array}{@{}rcl@{}} &&\frac{\partial(X_{j}{H_{j}^{T}})_{av}} {\partial X_{j,lk}}\\ &=&\frac{\partial\sum\limits^{\mathrm{n}_{j}}_{p=1}X_{j,ap}H_{j,pv}^{T}} {\partial X_{j,lk}}\\ &=&\sum\limits^{\mathrm{n}_{j}}_{p=1}\left( \frac{\partial X_{j,ap}H_{j,vp}} {\partial X_{j,lk}}\right)\\ &=&\sum\limits^{\mathrm{n}_{j}}_{p=1}\left( H_{j,vp}\frac{\partial X_{j,ap}} {\partial X_{j,lk}}\right)\\ &=&\sum\limits^{\mathrm{n}_{j}}_{p=1}\left( H_{j,vp}\delta_{la}\delta_{pk}\right)\\ &=&H_{j,vk}\delta_{la}. \end{array} $$
(62)

Similarly, the right fraction term in parentheses of Eq. 61becomes,

$$ \frac{\partial(H_{j}{X_{j}^{T}})_{ca}} {\partial X_{j,lk}} =H_{j,ck}\delta_{la}. $$
(63)

By substituting Eqs. 62 and 63 into Eq. 61 and using Lemma 6, we can obtain

$$\begin{array}{@{}rcl@{}} &&\!\!\!\!\!\sum\limits^{\mathrm{m}}_{a=1}\left( (H_{j}{X_{j}^{T}})_{ca}\frac{\partial(X_{j}{H_{j}^{T}})_{av}} {\partial X_{j,lk}} +(X_{j}{H_{j}^{T}})_{av}\frac{\partial(H_{j}{X_{j}^{T}})_{ca}} {\partial X_{j,lk}} \right)\\ &=&\sum\limits^{\mathrm{m}}_{a=1}\left( (H_{j}{X_{j}^{T}})_{ca}H_{j,vk}\delta_{la} +(X_{j}{H_{j}^{T}})_{av}H_{j,ck}\delta_{la} \right)\\ &=&\left( (H_{j}{X_{j}^{T}})_{cl}H_{j,vk} +(X_{j}{H_{j}^{T}})_{lv}H_{j,ck} \right). \end{array} $$
(64)

By substituting Eq. 64 into Eq. 60 and using Lemma 6, we have

$$\begin{array}{@{}rcl@{}} &&\sum\limits^{\mathrm{m}}_{v=1}\sum\limits^{\mathrm{m}}_{c=1} \left( \delta_{cm}\delta_{nv}\left( \frac{\partial\left( H_{j}{X_{j}^{T}}X_{j}{H_{j}^{T}}\right)_{cv}}{\partial X_{j,lk}}\right)\right)\\ &=&\sum\limits^{\mathrm{m}}_{v=1}\sum\limits^{\mathrm{m}}_{c=1} \left( \delta_{cm}\delta_{nv}\left( (H_{j}{X_{j}^{T}})_{cl}H_{j,vk} +(X_{j}{H_{j}^{T}})_{lv}H_{j,ck} \right)\right)\\ &=&\sum\limits^{\mathrm{m}}_{v=1} \left( \delta_{nv}\left( (H_{j}{X_{j}^{T}})_{ml}H_{j,vk} +(X_{j}{H_{j}^{T}})_{lv}H_{j,mk} \right)\right)\\ &=&\left( (H_{j}{X_{j}^{T}})_{ml}H_{j,nk} +(X_{j}{H_{j}^{T}})_{ln}H_{j,mk} \right). \end{array} $$
(65)

By substituting Eqs. 57 and 65 into Eq. 52, we have

$$\begin{array}{@{}rcl@{}} &&\frac{\partial F(X)}{\partial X_{j,lk}}\\ &=&\frac{1}{2}\sum\limits^{\mathrm{m}}_{m=1}\sum\limits^{\mathrm{m}}_{n=1}\frac{\partial\left( \log\left|\mathcal{G}\right|\right)} {\partial\left( \mathcal{G}\right)_{mn} }\frac{\partial\left( \mathcal{G}\right)_{mn}}{\partial X_{j,lk}}\\ &=&\frac{1}{2}\sum\limits^{\mathrm{m}}_{m=1}\sum\limits^{\mathrm{m}}_{n=1} \left( \left( \mathcal{G}\right)^{-1}\right)_{nm}\\ &&\times\left( (H_{j}{X_{j}^{T}})_{ml}H_{j,nk}+(X_{j}{H_{j}^{T}})_{ln}H_{j,mk}\right)\\ &=&\frac{1}{2}\sum\limits^{\mathrm{m}}_{m=1}\sum\limits^{\mathrm{m}}_{n=1} \left( \left( \mathcal{G}\right)^{-1}\right)_{nm} \left( (H_{j}{X_{j}^{T}})_{ml}H_{j,nk}\right)\\ &&+\frac{1}{2}\sum\limits^{\mathrm{m}}_{m=1}\sum\limits^{\mathrm{m}}_{n=1} \left( \left( \mathcal{G}\right)^{-1}\right)_{nm} \left( (X_{j}{H_{j}^{T}})_{ln}H_{j,mk}\right). \end{array} $$
(66)

Using Lemmas 5 and 6, and switching the places of entities such that the derivation can be written as [⋅] l k , wehave

$$\begin{array}{@{}rcl@{}} \frac{\partial F(X)}{\partial X_{j,lk}} &=&\frac{1}{2}\sum\limits^{\mathrm{m}}_{m=1} \left( {H_{j}^{T}}\left( \mathcal{G}\right)^{-1}\right)_{km} (H_{j}{X_{j}^{T}})_{ml}\\ &&+\frac{1}{2}\sum\limits^{\mathrm{m}}_{m=1} \left( (X_{j}{H_{j}^{T}})\left( \mathcal{G}\right)^{-1}\right)_{lm} H_{j,mk}\\ &=&\frac{1}{2} \left( {H_{j}^{T}}\left( \mathcal{G}\right)^{-1}H_{j}{X_{j}^{T}}\right)_{kl}\\ &&+\frac{1}{2} \left( (X_{j}{H_{j}^{T}})\left( \mathcal{G}\right)^{-1}H_{j}\right)_{lk}\\ &=&\frac{1}{2} \left( X_{j}{H^{T}_{j}}\left( \left( \mathcal{G}\right)^{-1}\right)^{T}H_{j}\right)_{lk}\\ &&+\frac{1}{2} \left( X_{j}{H^{T}_{j}}\left( \mathcal{G}\right)^{-1}H_{j}\right)_{lk}. \end{array} $$
(67)

Thus Lemma 7 is proved. □

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Yang, ZX., Zhao, GS., Li, G. et al. Matrix differentiation for capacity region of Gaussian multiple access channels under weighted total power constraint. Ann. Telecommun. 72, 703–715 (2017). https://doi.org/10.1007/s12243-017-0610-7

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