Skip to main content
SpringerLink
Log in

Energy-Efficient Design in Cognitive MIMO Systems

  • Living reference work entry
  • First Online:
Handbook of Cognitive Radio

  • 106 Accesses

Abstract

The energy-efficient design for TDMA (time-division multiple access) MIMO (multiple-input multiple-output) cognitive radio (CR) networks can be treated as the joint optimization over both the time resource and the transmit precoding matrices to minimize the overall energy consumption. Compared with the traditional MIMO networks, the challenge here is that the secondary users (SUs) may not be able to obtain the channel state information (CSI) to the primary receivers. The corresponding mathematical formulation turns out to be non-convex and thus of high complexity to solve in general. This chapter covers both the transmission choices for each SU: single-data-stream transmission and multiple-data-stream transmission. Fortunately, by applying a proper optimization decomposition, it can be shown that the optimal solution can be found in polynomial time in both cases. In practical wireless system, the time is usually allocated in the unit of slot. Moreover, by exploring the special structure of this particular problem, it can be shown that the optimal time slots allocation can be obtained in polynomial time with a simple greedy algorithm. Simulation results show that the energy-optimal transmission scheme adapts to the traffic load of the secondary system to create a win-win situation where the SUs are able to decrease the energy consumption and the PUs experience less interference from the secondary system. The effect is particularly pronounced when the secondary system is underutilized.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. Barzilai J, Borwein J (1988) Two-point step size gradient methods. IMA J Numer Anal 8(1):141–148

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont

    MATH  Google Scholar 

  3. Birgin E, Martínez J, Raydan M (2000) Nonmonotone spectral projected gradient methods on convex sets. SIAM J Optim 10(4):1196–1211

    Article  MathSciNet  MATH  Google Scholar 

  4. Blum RS (2003) MIMO capacity with interference. IEEE J Sel Areas Commun 21(5):793–801

    Article  Google Scholar 

  5. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  6. Cui S, Goldsmith AJ, Bahai A (2004) Energy-efficiency of MIMO and cooperative MIMO techniques in sensor networks. IEEE J Sel Areas Commun 22(6):1089–1098

    Article  Google Scholar 

  7. Fu L, Kim H, Huang J, Liew SC, Chiang M (2011) Energy conservation and interference mitigation: from decoupling property to win-win strategy. IEEE Trans Wirel Commun 10(11):3943–3955

    Article  Google Scholar 

  8. Gershman AB, Sidiropoulos ND, Shahbazpanahi S, Bengtsson M, Ottersten B (2010) Convex optimization-based beamforming: from receive to transmit and network designs. IEEE Signal Process Mag 27(3):62–75

    Article  Google Scholar 

  9. Gesbert D, Shafi M, shan Shiu D, Smith PJ, Naguib A (2003) From theory to practice: an overview of MIMO space-time coded wireless systems. IEEE J Sel Areas Commun 21(3):281–302

    Google Scholar 

  10. Grippo L, Lampariello F, Lucidi S (1986) A nonmonotone line search technique for Newton’s method. SIAM J Numer Anal 23(4):707–716

    Article  MathSciNet  MATH  Google Scholar 

  11. Haykin S (2005) Cognitive radio: brain-empowered wireless communications. IEEE J Sel Areas Commun 23(2):201–220

    Article  Google Scholar 

  12. Huang Y, Li Q, Ma WK, Zhang S (2012) Robust multicast beamforming for spectrum sharing-based cognitive radios. IEEE Trans Signal Process 60(1):527–533

    Article  MathSciNet  Google Scholar 

  13. Ibaraki T, Katoh N (1988) Resource allocation problems: algorithmic approaches. MIT Press, Cambridge

    MATH  Google Scholar 

  14. Johnson DH, Dudgeon DE (1993) Array signal processing: concepts and techniques. Prentice Hall signal processing series. P T R Prentice-Hall Inc., Englewood Cliffs

    MATH  Google Scholar 

  15. Kim H, Chae CB, de Veciana G, Heath RW Jr (2009) A cross-layer approach to energy efficiency for adaptive MIMO systems exploiting spare capacity. IEEE Trans Wirel Commun 8(8):4264–4275

    Article  Google Scholar 

  16. Kobayashi M, Caire G (2007) Joint beamforming and scheduling for a multi-antenna downlink with imperfect transmitter channel knowledge. IEEE J Sel Areas Commun 25(7):1468–1477

    Article  Google Scholar 

  17. Liang YC, Chen KC, Li GY, Mähönen P (2011) Cognitive radio networking and communications: an overview. IEEE Trans Veh Technol 60(7):3386–3407

    Article  Google Scholar 

  18. Ma J, Zhang YJ, Su X, Yao Y (2008) On capacity of wireless ad hoc networks with MIMO MMSE receivers. IEEE Trans Wirel Commun 7(12):5493–5503

    Article  Google Scholar 

  19. Palomar DP, Fonollosa JR (2005) Practical algorithms for a family of waterfilling solutions. IEEE Trans Signal Process 53(2):686–695

    Article  MathSciNet  Google Scholar 

  20. Phan KT, Vorobyov SA, Sidiropoulos ND, Tellambura C (2009) Spectrum sharing in wireless networks via QoS-aware secondary multicast beamforming. IEEE Trans Signal Process 57(6):2323–2335

    Article  MathSciNet  Google Scholar 

  21. Rappaport TS (2002) Wireless communications: principles and practice, 2nd edn. Prentice Hall PTR, Upper Saddle River

    MATH  Google Scholar 

  22. Razaviyayn M, Sanjabi M, Luo ZQ (2012) Linear transceiver design for interference alignment: complexity and computation. IEEE Trans Inf Theory 58(5):2896–2910

    Article  MathSciNet  Google Scholar 

  23. Schmidt M (2010) Graphical model structure learning with l 1-regularization. Ph.D. thesis, University of British Columbia

    Google Scholar 

  24. Sidiropoulos ND, Davidson TN, Luo ZQ (2006) Transmit beamforming for physical-layer multicasting. IEEE Trans Signal Process 54(6):2239–2251

    Article  Google Scholar 

  25. Telatar E (1995) Capacity of multi-antenna gaussian channels. AT&T Bell Laboratories, Tech. Memo

    Google Scholar 

  26. Uysal-Biyikoglu E, Prabhakar B, Gamal AE (2002) Energy-efficient packet transmission over a wireless link. IEEE ACM Trans Netw 10(4):487–499

    Article  Google Scholar 

  27. Yu W, Lan T (2007) Transmitter optimization for the multi-antenna downlink with per-antenna power constraints. IEEE Trans Signal Process 55(6):2646–2660

    Article  MathSciNet  Google Scholar 

  28. Zhang G, Ma S, Wong KK, Ng TS (2010) Robust beamforming in cognitive radio. IEEE Trans Wirel Commun 9(2):570–576

    Article  Google Scholar 

  29. Zhang L, Liang YC, Xin Y (2008) Joint beamforming and power allocation for multiple access channels in cognitive radio networks. IEEE J Sel Areas Commun 26(1):38–51

    Article  Google Scholar 

  30. Zhang L, Xin Y, Liang YC (2009) Weighted sum rate optimization for cognitive radio MIMO broadcast channels. IEEE Trans Wirel Commun 8(6):2950–2959

    Article  Google Scholar 

  31. Zhang R, Liang YC (2008) Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks. IEEE J Sel Topics Signal Process 2(1):88–102

    Article  Google Scholar 

  32. Zhang R, Liang YC, Cui S (2010) Dynamic resource allocation in cognitive radio networks: a convex optimization perspective. IEEE Signal Process Mag 27(3):102–114

    Article  Google Scholar 

  33. Zhang YJ, So AMC (2011) Optimal spectrum sharing in MIMO cognitive radio networks via semidefinite programming. IEEE J Sel Areas Commun 29(2):362–373

    Article  Google Scholar 

  34. Zheng G, Wong KK, Ottersten B (2009) Robust cognitive beamforming with bounded channel uncertainties. IEEE Trans Signal Process 57(12):4871–4881

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Communication Engineering, Xiamen University, Room 705B, Administration Building C, Haiyun Park, 361005, Xiamen, China

    Liqun Fu

Authors
  1. Liqun Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liqun Fu .

Editor information

Editors and Affiliations

  1. Electrical Engg Bldg, Rm 325, The University of New South Wales, Sydney, New South Wales, Australia

    Wei Zhang

Section Editor information

  1. Department of Information Engineering, The Chinese University of Hong Kong, Room 834, Ho Sin Hang Engineering Building, Shatin, New Territory, Hong Kong, China SAR

    Jianwei Huang

  2. School of Data and Computer Science, Sun Yat-sen University, 510006, Guangzhou, Guangdong, 中国

    Xu Chen

Appendix

Appendix

Proof of Proposition 1:

We first show that \(E_{S_{k}}\left (t_{S_{k}}\right )\) is a continuous function in \(t_{S_{k}}\) by showing \(\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{-}\left (m_{S_{k}}\right )}E_{S_{k}}\left (t_{S_{k}}\right ) =\lim \limits _{t_{S_{k}}\rightarrow \tau _{S_{k}}^{+}\left (m_{S_{k}}\right )}E_{S_{k}}\left (t_{S_{k}}\right )\):

$$\displaystyle{ \begin{array}{ll} &\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{-}\left (m_{S_{k}}\right )}E_{S_{k}}\left (t_{S_{k}}\right ) =\tau _{S_{k}}\left (m_{S_{k}}\right )\left (\left (m_{S_{k}} + 1\right )\left ( \frac{\exp \left ( \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )} \right )} {\prod \limits _{i=1}^{m_{S_{k}}+1 }\lambda _{S_{k},i}}\right )^{ \frac{1} {m_{S_{k}}+1} } -\sum \limits _{i=1}^{m_{S_{k}}+1} \frac{1} {\lambda _{S_{k},i}}\right ) \\ & =\,\tau _{S_{k}}\left (m_{S_{k}}\right )\left (\!\left (m_{S_{k}}+1\right )\left (\frac{\exp \left (\left (\sum \limits _{i=1}^{m_{S_{k}} }\log \lambda _{S_{k},i}\right )-m_{S_{k}}\log \lambda _{S_{k},\left (m_{S_{ k}}+1\right )}\right )} {\prod \limits _{i=1}^{m_{S_{k}}+1 }\lambda _{S_{k},i}} \right )^{ \frac{1} {m_{S_{k}}+1} }-\sum \limits _{i=1}^{m_{S_{k}}+1} \frac{1} {\lambda _{S_{k},i}}\right ) \\ & =\tau _{S_{k}}\left (m_{S_{k}}\right )\left (\left (m_{S_{k}} + 1\right )\left ( \frac{\prod \limits _{i=1}^{m_{S_{k}} }\lambda _{S_{k},i}} {\lambda _{S_{k},\left (m_{S_{ k}}+1\right )}^{m_{S_{k}} }\prod \limits _{i=1}^{m_{S_{k}}+1 }\lambda _{S_{k},i}}\right )^{ \frac{1} {m_{S_{k}}+1} } -\sum \limits _{i=1}^{m_{S_{k}}+1} \frac{1} {\lambda _{S_{k},i}}\right ) \\ & =\tau _{S_{k}}\left (m_{S_{k}}\right )\left ( \frac{m_{S_{k}}} {\lambda _{S_{k},\left (m_{S_{ k}}+1\right )}} -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}}\right ), \end{array} }$$

and

$$\displaystyle{ \begin{array}{ll} &\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{+}\left (m_{S_{k}}\right )}E_{S_{k}}\left (t_{S_{k}}\right ) =\tau _{S_{k}}\left (m_{S_{k}}\right )\left (m_{S_{k}}\left (\frac{\exp \left ( \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )} \right )} {\prod \limits _{i=1}^{m_{S_{k}} }\lambda _{S_{k},i}} \right )^{ \frac{1} {m_{S_{k}}} } -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}}\right ) \\ & =\tau _{S_{k}}\left (m_{S_{k}}\right )\left (m_{S_{k}}\left (\frac{\exp \left (\left (\sum \limits _{i=1}^{m_{S_{k}} }\log \lambda _{S_{k},i}\right )-m_{S_{k}}\log \lambda _{S_{k},\left (m_{S_{ k}}+1\right )}\right )} {\prod \limits _{i=1}^{m_{S_{k}} }\lambda _{S_{k},i}} \right )^{ \frac{1} {m_{S_{k}}} } -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}}\right ) \\ & =\tau _{S_{k}}\left (m_{S_{k}}\right )\left ( \frac{m_{S_{k}}} {\lambda _{S_{k},\left (m_{S_{ k}}+1\right )}} -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}}\right ). \end{array} }$$

Thus we have

$$\displaystyle{\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{-}\left (m_{S_{k}}\right )}E_{S_{k}}\left (t_{S_{k}}\right ) =\lim \limits _{t_{S_{k}}\rightarrow \tau _{S_{k}}^{+}\left (m_{S_{k}}\right )}E_{S_{k}}\left (t_{S_{k}}\right ).}$$

Therefore, \(E_{S_{k}}\left (t_{S_{k}}\right )\) is a continuous function in \(t_{S_{k}}\).

Next, we show that \(\frac{dE_{S_{k}}} {dt_{S_{k}}}\) is a continuous function in \(t_{S_{k}}\) by showing \(\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{-}\left (m_{S_{k}}\right )}\frac{dE_{S_{k}}} {dt_{S_{k}}} =\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{+}\left (m_{S_{k}}\right )}\frac{dE_{S_{k}}} {dt_{S_{k}}}\). The first-order derivative of \(E_{S_{k}}\left (t_{S_{k}}\right )\) with respect to \(t_{S_{k}}\) is

$$\displaystyle{ \frac{dE_{S_{k}}} {dt_{S_{k}}} = \left (D_{S_{k}}^{{\ast}}\left (t_{ S_{k}}\right ) - \frac{R_{S_{k}}} {wt_{S_{k}}}\right )\left ( \frac{\exp \left ( \frac{R_{S_{k}}} {wt_{S_{k}}}\right )} {\prod \limits _{i=1}^{D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )}\lambda _{ S_{k},i}}\right )^{ \frac{1} {D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )} } -\sum \limits _{i=1}^{D_{S_{k}}^{{\ast}}\left (t_{ S_{k}}\right )} \frac{1} {\lambda _{S_{k},i}}. }$$

It can be shown that

$$\displaystyle{ \begin{array}{ll} &\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{-}\left (m_{S_{k}}\right )}\frac{dE_{S_{k}}} {dt_{S_{k}}} = \left (\left (m_{S_{k}} + 1\right ) - \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )}\right )\left ( \frac{\exp \left ( \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )} \right )} {\prod \limits _{i=1}^{m_{S_{k}}+1 }\lambda _{S_{k},i}}\right )^{ \frac{1} {m_{S_{k}}+1} } -\sum \limits _{i=1}^{m_{S_{k}}+1} \frac{1} {\lambda _{S_{k},i}} \\ & = \left (\left (m_{S_{k}} + 1\right ) - \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )}\right )\left (\frac{\exp \left (\left (\sum \limits _{i=1}^{m_{S_{k}} }\log \lambda _{S_{k},i}\right )-m_{S_{k}}\log \lambda _{S_{k},\left (m_{S_{ k}}+1\right )}\right )} {\prod \limits _{i=1}^{m_{S_{k}}+1 }\lambda _{S_{k},i}} \right )^{ \frac{1} {m_{S_{k}}+1} } \\ & \qquad -\sum \limits _{i=1}^{m_{S_{k}}+1} \frac{1} {\lambda _{S_{k},i}} \\ & = \left (\left (m_{S_{k}} + 1\right ) - \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )}\right ) \frac{1} {\lambda _{S_{k},\left (m_{S_{ k}}+1\right )}} -\sum \limits _{i=1}^{m_{S_{k}}+1} \frac{1} {\lambda _{S_{k},i}} \\ & = \left (m_{S_{k}} - \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )}\right ) \frac{1} {\lambda _{S_{k},\left (m_{S_{ k}}+1\right )}} -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}}, \end{array} }$$

and

$$\displaystyle{ \begin{array}{ll} &\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{+}\left (m_{S_{k}}\right )}\frac{dE_{S_{k}}} {dt_{S_{k}}} = \left (m_{S_{k}} - \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )}\right )\left (\frac{\exp \left ( \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )} \right )} {\prod \limits _{i=1}^{m_{S_{k}} }\lambda _{S_{k},i}} \right )^{ \frac{1} {m_{S_{k}}} } -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}} \\ & = \left (m_{S_{k}} - \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )}\right )\left (\frac{\exp \left (\left (\sum \limits _{i=1}^{m_{S_{k}} }\log \lambda _{S_{k},i}\right )-m_{S_{k}}\log \lambda _{S_{k},\left (m_{S_{ k}}+1\right )}\right )} {\prod \limits _{i=1}^{m_{S_{k}} }\lambda _{S_{k},i}} \right )^{ \frac{1} {m_{S_{k}}} } -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}} \\ & = \left (m_{S_{k}} - \frac{R_{S_{k}}} {w\tau _{S_{k}}\left (m_{S_{k}}\right )}\right ) \frac{1} {\lambda _{S_{k},\left (m_{S_{ k}}+1\right )}} -\sum \limits _{i=1}^{m_{S_{k}} } \frac{1} {\lambda _{S_{k},i}}. \end{array} }$$

Thus, we have

$$\displaystyle{\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{-}\left (m_{S_{k}}\right )}\frac{dE_{S_{k}}} {dt_{S_{k}}} =\lim \limits _{t_{S_{ k}}\rightarrow \tau _{S_{k}}^{+}\left (m_{S_{k}}\right )}\frac{dE_{S_{k}}} {dt_{S_{k}}}.}$$

Therefore, \(\frac{dE_{S_{k}}} {dt_{S_{k}}}\) is a continuous function in \(t_{S_{k}}\).

The second-order derivative of \(E_{S_{k}}\left (t_{S_{k}}\right )\) with respect to \(t_{S_{k}}\) is

$$\displaystyle{ \frac{d^{2}E_{S_{k}}} {dt_{S_{k}}^{2}} = \frac{R_{S_{k}}^{2}} {w^{2}t_{S_{k}}^{3}D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )}\left ( \frac{\exp \left ( \frac{R_{S_{k}}} {wt_{S_{k}}}\right )} {\prod \limits _{i=1}^{D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )}\lambda _{ S_{k},i}}\right )^{ \frac{1} {D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )} }, }$$

which is always positive for any positive \(t_{S_{k}}\). However, it is not continuous in \(t_{S_{k}}\), with the noncontinuous points at \(t_{S_{k}} =\tau _{S_{k}}\left (m_{S_{k}}\right )\), (\(m_{S_{k}} = \left \{1,\cdots \,,\left (W_{S_{k}} - 1\right )\right \}\)).

Next we show that \(\frac{dE_{S_{k}}} {dt_{S_{k}}}\) is always negative for any positive \(t_{S_{k}}\). Since \(\frac{d^{2}E_{ S_{k}}} {dt_{S_{k}}^{2}}\) is always positive, this means that \(\frac{dE_{S_{k}}} {dt_{S_{k}}}\) is an increasing function in \(t_{S_{k}}\). Thus, we have

$$\displaystyle{ \begin{array}{ll} &\frac{dE_{S_{k}}} {dt_{S_{k}}} <\lim \limits _{t_{S_{ k}}\rightarrow \infty }\left (\left (D_{S_{k}}^{{\ast}}\left (t_{S_{ k}}\right ) - \frac{R_{S_{k}}} {wt_{S_{k}}}\right )\left ( \frac{\exp \left ( \frac{R_{S_{k}}} {wt_{S_{k}}}\right )} {\prod \limits _{i=1}^{D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right ) }\lambda _{S_{k},i}}\right )^{ \frac{1} {D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )} } -\sum \limits _{i=1}^{D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )} \frac{1} {\lambda _{S_{k},i}}\right ) \\ & = \left.\left (D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )\left ( \frac{1} {\prod \limits _{i=1}^{D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right ) }\lambda _{S_{k},i}}\right )^{ \frac{1} {D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )} } -\sum \limits _{i=1}^{D_{S_{k}}^{{\ast}}\left (t_{S_{k}}\right )} \frac{1} {\lambda _{S_{k},i}}\right )\right \vert _{D_{S_{ k}}^{{\ast}}\left (t_{S_{k}}\right )=1} = \frac{1} {\lambda _{S_{k},1}} - \frac{1} {\lambda _{S_{k},1}} = 0. \end{array} }$$
(32)

The second line of (32) follows from (28), which states that when \(t_{S_{k}}\) approaches infinity, the optimal number of data streams equals 1. Therefore, we can find that the optimal energy consumption \(E_{S_{k}}\left (t_{S_{k}}\right )\) is a strictly convex, continuous, first-order differentiable, and monotonically decreasing function in \(t_{S_{k}}\). □

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this entry

Cite this entry

Fu, L. (2017). Energy-Efficient Design in Cognitive MIMO Systems. In: Zhang, W. (eds) Handbook of Cognitive Radio . Springer, Singapore. https://doi.org/10.1007/978-981-10-1389-8_26-1

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-1389-8_26-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-1389-8

  • Online ISBN: 978-981-10-1389-8

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation