宽带无线网络分时间尺度资源分配与非凸优化
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摘要
近年来，在宽带无线网络中向多媒体应用提供服务质量(QoS)保证已成为研究热点。其中，无线资源的分配与调度是提供QoS保证的关键，当前的研究集中在针对无线信道容量的时变和位置依赖特性，设计算法来有效利用宝贵的无线资源，支持用户的QoS要求，同时满足一定的公平性。其中有大量问题尚待解决，有的概念也没有统一的认识。以公平性为例，相关文献中采用的有：占有资源的公平性[1]、吞吐量公平性[2]、收益Utility公平性[3]和价格Price公平性[4][5]等。
本文提出了分时间尺度(Time-scale Decomposition)的无线资源分配与调度策略和系统模型。（1）针对无线物理层，引入效率函数来表征在一定残余误码率BER要求和一定信道信噪比SNR情况下，上层应用正确传输的数据量和下层无线资源的关系。（2）针对多媒体应用，使用不同的Utility函数来表征不同的QoS要求，其优点有：一是可以同时表征业务对带宽和时延的要求；二是，可以表征某些应用的“软”QoS要求。（3）针对无线信道衰落的多时间尺度特性，将信道条件SNR分解到不同时间尺度上，在不同时间尺度上针对不同的衰落特性来设计不同的算法，完成不同的任务：在大时间尺度（帧）上进行资源分配，面向应用提供QoS保证，保持应用间Price公平性的情况下，寻求系统总收益Utility的最大化；在小时间尺度（时隙）上进行时隙调度，利用各用户信道容量的时变性，使各用户获得比平均信道条件情况下更高的吞吐量。
该模型中的资源分配问题是一个非凸的非线性优化问题。论文定义了系统价格p的特征资源分配向量，提出使用该向量来求得最优解或次优解的算法，并给出了该算法求得最优解的充分条件。论文证明了：当该条件不满足时，次优解与全局最优解的距离（用百分比表示）不大于次优解中未分配的带宽占总数的百分比，不大于单个应用在临界状态时获得的无线资源占系统总量的百分比。
最后，论文对该模型中的资源分配和资源调度分别提出了各自的低复杂度在线求解算法。计算机仿真结果表明该资源分配在线求解算法收敛于全局最优解，也说明了分时间尺度的资源分配和调度策略能够在提供QoS保证的同时，相对于纯调度策略，提高系统性能50%以上。
In resent years, the research on how to provide quality of service (QoS) guarantees to multimedia applications in broadband wireless networks has been a hot spot. Radio resource allocation and scheduling is one of key functions to provide QoS guarantees in wireless networks. Current research focuses on design of efficient algorithms taking into account the special characteristics of the wireless environment such as time-varying channel capacity and location-dependent errors. These algorithms shall maximize the utilization of the wireless channels and guarantee QoS for the users, while providing certain fairness between users. Many questions remain open and even some definitions are ambiguous in this area. For example, there are several different definitions of fairness addressed in related works, such as time-fraction fairness [1], throughput fairness [2], utility fairness and price fairness [4][5].
In this paper, a novel scheme of radio resource allocation and scheduling and its system model is proposed through the time-scale decomposition approach. (1) For wireless physical layers, the efficiency function is defined to quantify the upper layer throughput per unit of wireless resources while maintaining a certain maximum remnant bit error rate subjected to a certain SNR. (2) For multimedia applications, it is advantageous to use the different types of utility functions to express different type of QoS requirements, because it can express two major QoS metrics (bandwidth and delay) simultaneously and be used to represent adaptive QoS requirement. (3) For the wireless channels subject to several types of fading existing in different time scales, the dynamics of channel conditions are decoupled into two random processes with different mathematic properties in different time scales. Two algorithms in this scheme are proposed to dealing with each time scale: the resource optimizer allocates the resource to maximize the total revenue with price fairness and provide QoS guarantees to applications, and the slot scheduler exploits the time variability of
channel capacities to provide higher throughputs to users than the ones obtained in mean channel conditions.
The problem of resource allocation in this scheme is a non-convex non-linear optimization problem. By defining an eigen allocation vector, we propose an algorithm to obtain the global optima or sub-optimal solution. The sufficient condition to obtain the global optima is also presented. In this paper, it is proved that when the condition is not satisfied, the gap between the sub-optima solution and the global optima (expressed by a percentage of the global optima) is no more than the ratio between the left resources of the sub-optimal solution and the total resources, and is no more than the ratio between the cutoff resources obtained by one application and the total resources.
Finally, the low-complexity online algorithms are presented for the radio resource allocation and scheduling in this scheme. The results of the computer simulations show that the online resource-allocation algorithm converges to the globe optima. The results also show that our scheme can obtain a 50% gain over the Only Scheduler scheme, while providing QoS guarantees.
本文提出了分时间尺度(Time-scale Decomposition)的无线资源分配与调度策略和系统模型。（1）针对无线物理层，引入效率函数来表征在一定残余误码率BER要求和一定信道信噪比SNR情况下，上层应用正确传输的数据量和下层无线资源的关系。（2）针对多媒体应用，使用不同的Utility函数来表征不同的QoS要求，其优点有：一是可以同时表征业务对带宽和时延的要求；二是，可以表征某些应用的“软”QoS要求。（3）针对无线信道衰落的多时间尺度特性，将信道条件SNR分解到不同时间尺度上，在不同时间尺度上针对不同的衰落特性来设计不同的算法，完成不同的任务：在大时间尺度（帧）上进行资源分配，面向应用提供QoS保证，保持应用间Price公平性的情况下，寻求系统总收益Utility的最大化；在小时间尺度（时隙）上进行时隙调度，利用各用户信道容量的时变性，使各用户获得比平均信道条件情况下更高的吞吐量。
该模型中的资源分配问题是一个非凸的非线性优化问题。论文定义了系统价格p的特征资源分配向量，提出使用该向量来求得最优解或次优解的算法，并给出了该算法求得最优解的充分条件。论文证明了：当该条件不满足时，次优解与全局最优解的距离（用百分比表示）不大于次优解中未分配的带宽占总数的百分比，不大于单个应用在临界状态时获得的无线资源占系统总量的百分比。
最后，论文对该模型中的资源分配和资源调度分别提出了各自的低复杂度在线求解算法。计算机仿真结果表明该资源分配在线求解算法收敛于全局最优解，也说明了分时间尺度的资源分配和调度策略能够在提供QoS保证的同时，相对于纯调度策略，提高系统性能50%以上。
In resent years, the research on how to provide quality of service (QoS) guarantees to multimedia applications in broadband wireless networks has been a hot spot. Radio resource allocation and scheduling is one of key functions to provide QoS guarantees in wireless networks. Current research focuses on design of efficient algorithms taking into account the special characteristics of the wireless environment such as time-varying channel capacity and location-dependent errors. These algorithms shall maximize the utilization of the wireless channels and guarantee QoS for the users, while providing certain fairness between users. Many questions remain open and even some definitions are ambiguous in this area. For example, there are several different definitions of fairness addressed in related works, such as time-fraction fairness [1], throughput fairness [2], utility fairness and price fairness [4][5].
In this paper, a novel scheme of radio resource allocation and scheduling and its system model is proposed through the time-scale decomposition approach. (1) For wireless physical layers, the efficiency function is defined to quantify the upper layer throughput per unit of wireless resources while maintaining a certain maximum remnant bit error rate subjected to a certain SNR. (2) For multimedia applications, it is advantageous to use the different types of utility functions to express different type of QoS requirements, because it can express two major QoS metrics (bandwidth and delay) simultaneously and be used to represent adaptive QoS requirement. (3) For the wireless channels subject to several types of fading existing in different time scales, the dynamics of channel conditions are decoupled into two random processes with different mathematic properties in different time scales. Two algorithms in this scheme are proposed to dealing with each time scale: the resource optimizer allocates the resource to maximize the total revenue with price fairness and provide QoS guarantees to applications, and the slot scheduler exploits the time variability of
channel capacities to provide higher throughputs to users than the ones obtained in mean channel conditions.
The problem of resource allocation in this scheme is a non-convex non-linear optimization problem. By defining an eigen allocation vector, we propose an algorithm to obtain the global optima or sub-optimal solution. The sufficient condition to obtain the global optima is also presented. In this paper, it is proved that when the condition is not satisfied, the gap between the sub-optima solution and the global optima (expressed by a percentage of the global optima) is no more than the ratio between the left resources of the sub-optimal solution and the total resources, and is no more than the ratio between the cutoff resources obtained by one application and the total resources.
Finally, the low-complexity online algorithms are presented for the radio resource allocation and scheduling in this scheme. The results of the computer simulations show that the online resource-allocation algorithm converges to the globe optima. The results also show that our scheme can obtain a 50% gain over the Only Scheduler scheme, while providing QoS guarantees.
引文
[1] Xin Liu, Edwin K. P. Chong, and Ness B. Shroff, "Transmission scheduling for efficient wireless network utilization," in Proc. INFOCOM'01, July 2001, pp. 776-785.
[2] Sem Borst and Phil Whiting, "Dynamic Rate Control Algorithms for HDR Throught Optimization" in Proc. INFOCOM'01, July 2001.
[3] R.R.-F. Liao, and A.T. Campbell, "A Utility-Based Approach for Quantitative Adaptation in Wireless Packet Networks", ACM Journal on Wireless Networks (WINET), Vol. 7, No. 5, Sept. 2001, pp. 541-557.
[4] M. Xiao, N. B. Shroff, and E. K. P. Chong, "Utility-based power control in cellular wireless systems", in Proc INFOCOM'01, July 2000.
[5] Hao Lin, Wei Wu, Yong Ren and Xiuming Shan, "A Time-scale Decomposition Approach to Optimize Wireless Packet Resource Allocation and Scheduling", IEEE Wireless communications and Networking Conference (WCNC) 2002, Orlando, FL, USA, pp. 699-706.
[6] "The Book of Visions 2001 - Visions of the Wireless World," Wireless World Research Forum; http://www.wireless-world-research.org
[7] Johan De Vriendt, Philippe Lainé, Christophe Lerouge, and Xiaofeng Xu, Alcatel, "Mobile Network Evolution: A Revolution on the Move", IEEE Communications Magazine, vol.40, no.4, Apr. 2002, pp. 104-111.
[8] Yaxin Cao and Victor O. K. Li, "Scheduling Algorithms in Broad-Band Wireless Networks", Proceedings of IEEE, vol.89, no.1, Jan. 2001, pp. 76-87.
[9] P. Bhagwat, A. Krishna, and S. Tripathi, "Enhancing throughput over wireless LAN's using channel state dependent packet scheduling," in Proc. INFOCOM'96, Mar. 1996, pp. 1133-1140.
[10] C. Fragouli, V. Sivaraman, and M. Srivastava, "Controlled multimedia wireless link sharing via enhanced class-based queuing with channel-state dependent packet scheduling," in Proc. INFOCOM'98, vol. 2, Mar. 1998, pp. 572-580.
[11] S. Lu and V. Bharghavan, "Fair scheduling in wireless packet networks," IEEE/ACM Trans. Networking, vol. 7, no. 4, 1999, pp. 473-489.
[12] L. Kleinrock, Queueing Systems, vol. 2: Computer Applications. New York: Wiley, 1976.
[13] A. Demers, S. Keshav, and S. Shenker, "Analysis and simulation of a fair queueing algorithm," in Proc. ACM SIGCOMM'89, 1989, pp. 3-12.
[14] T. S. Eugene Ng, I. Stoica, and H. Zhang, "Packet fair queueing algorithms for wireless networks with location-dependent errors," in Proc. INFOCOM98, Mar. 1998, pp. 1103-1111.
[15] P. Ramanathan and P. Agrawal, "Adapting packet fair queueing algorithms to wireless networks," in ACM/IEEE MOBICOM'98, Dallas, TX, pp. 1-9.
[16] J. Gomez, A. T. Campbell, and H. Morikawa, "The Havana framework for supporting application and channel dependent QoS in wireless networks," in Proc. ICNP'99, Nov. 1999, pp. 235-244.
[17] M. Shreedhar and G. Varghese, "Efficient fair queueing using deficit round robin," in Proc. ACMSIGCOMM'95, Berkeley, CA, 1995, pp. 231-242.
[18] Paul Bender et al, "CDMA/HDR: A Bandwidth-Efficient High-Speed Wireless Data Service for Nomadic Users", IEEE Communication Magazine, vol. 38, no.6, July 2000, pp. 70-77.
[19] A. Jalali, R. Padovani and R. Pankaj, "Data Throughput of CDMA-HDR a High Efficiency-High Data Rate Personal Communication Wireless Systems", IEEE VTC, 2000.
[20] F. Kelly, "Charging and rate control for elastic traffic," European Transactions on Telecommunications, vol. 8, pp. 33-37, 1997.
[21] Sanjay Shakkottai and Alexander Stolyar, "Scheduling for Multiple flows Sharing a Time-Varying Channel: The Exponential Rule", Translations of the AMS, A volume in memory of F. Karpelevich, Providence, R.I.: American Mathematical Society, 2001.
[22] Randy H. Katz, "Adaptation and Mobility in Wireless Information Systems", IEEE Personal Communications Magazine, vol.1, no.1, First Quarter，1994.
[23] Theodore S. Rappaport, "Wireless Communications Principles and Practice", Prentice Hall Inc., 1996, pp. 139-298. (中译本：蔡涛, 李旭, 杜振民. "无线通信原理与应用", 北京, 电子工业出版社, 1999, pp. 50-145)
[24] M. Gudmundson, "Correlation model for shadow fading in mobile radio systems", IEE Electronics Letters, vol. 27, Nov 1991, pp. 2145-2146.
[25] R. H. Clarke, "A Statistical Theory of Mobile-radio Reception", Bell Systems Technical Journal, Vol. 47, 1968, pp. 957-1000.
[26] B.D. Fritchman, "A Binary Channel Characterization Using Partitioned Markov Chains", IEEE Transactions on Information Theory, Vol. 13, No. 2, Apr. 1967, pp. 221-227.
[27] S. Tsai, "Markov Characterization of the HF Channel", IEEE Transactions on Communication Technology, Vol. 17, No. 1, Feb. 1969, pp. 24-32.
[28] H.S. Wang, and N. Moayeri, "Finite-State Markov Channel-A Useful Model for Radio Communication Channels", IEEE Transactions on Vehicular Technology, Vol. 44, No. 1, Nov. 1995, pp. 163-171.
[29] M. Zorzi, R.R. Rao, and L.B. Milstein, "On the Accuracy of a First-Order Markov Model for Data Transmission on Fading Channels", Proc. IEEE ICUPC'95, Nov. 1995, pp. 211-215.
[30] H. Wang and N. Moayeri, "Finite state Markov channel-A useful model for radio communication channels," IEEE Trans. Veh. Technol., Feb. 1995, pp. 163-171.
[31] Charles Chien et al, "Adaptive Radio for Multimedia Wireless Links", IEEE Journal on Selected Areas in Communications, vol.17, no.5, May 1999, pp. 793-813.
[32] ISO/IEC 14496-1, "Information Technology-Coding of Audio-visual Objects, Part 1: Systems", ISO/IEC JT1/SC 29/WG 11 Draft International Standard, Dec. 1998.
[33] 3GPP, "Universal Mobile Telecommunication System (UMTS); QoS Concept and Architecture," TS 23.107 v.3.5.0, http://www.3gpp.org, 1999.
[34] Mohamed N. Moustafa, Ibrahim Habib, Mahmoud Naghshineh and Mohsen Guizani, "QoS-Enabled Broadband Mobile Access to Wireline Networks", IEEE Communications Maganize, vol. 40, no. 4, Apr 2002, pp. 50-56.
[35] D. I. Birx, "Chaotic oscillator and CMFFNSfor signal detection in noise environments", IEEE International Joint Conference on Neural Networks, 1992, 22: 821-888.
[36] 王冠宇，陶国良，陈行，等. 混沌振子在强噪声背景信号检测中的应用, 仪器仪表学报, 1997, 18(2): 209-212.
[37] 王冠宇，陈大军，林建亚，等. Duffing 振子微弱信号检测方法的统计特性研究. 电子学报. 1998，26(10): 38-43.
[38] Xun-he Yin, Ru-peng Feng, Yong Ren, "A kind of new amplifier". Chinese Physics, 2000, 9(2): 97-99.
[39] 童勤业，严筱刚，孔军，薛宗琪. “混沌”理论在测量中的应用. 电子科学学刊，1999，21(1): 42-49.
[40] Suqin Chen, Qinye Tong, Ruiyue Yuan. "N-nary A/D conversion with nonlinear dynamics method". Proceedings of the 1999 3 rd International Symposium on Test and Measurement. Xi'an, China: Chinese Society of Modern Technical Equipments: 1999, 193-194.
[41] 陈素琴，袁瑞跃，童勤业. 多进制式混沌A/D变换器研究. 计量学报，2000， 21(1): 12-16.
[42] A. Yilmaz, K.C. Wong, K.S. Chao. "A switched-current cyclic A/D conversion technique". Proceedings of the 37th Midwest Symposium on Circuits and Systems, IEEE, 1994. v2: 1155-1158.
[43] Mikael Gustavsson, J. Jacob Wikner, Nianxiong Nick Tan. CMOS Data Converters for communications, Boston: Kluwer Academic Publishers, 2000.
[2] Sem Borst and Phil Whiting, "Dynamic Rate Control Algorithms for HDR Throught Optimization" in Proc. INFOCOM'01, July 2001.
[3] R.R.-F. Liao, and A.T. Campbell, "A Utility-Based Approach for Quantitative Adaptation in Wireless Packet Networks", ACM Journal on Wireless Networks (WINET), Vol. 7, No. 5, Sept. 2001, pp. 541-557.
[4] M. Xiao, N. B. Shroff, and E. K. P. Chong, "Utility-based power control in cellular wireless systems", in Proc INFOCOM'01, July 2000.
[5] Hao Lin, Wei Wu, Yong Ren and Xiuming Shan, "A Time-scale Decomposition Approach to Optimize Wireless Packet Resource Allocation and Scheduling", IEEE Wireless communications and Networking Conference (WCNC) 2002, Orlando, FL, USA, pp. 699-706.
[6] "The Book of Visions 2001 - Visions of the Wireless World," Wireless World Research Forum; http://www.wireless-world-research.org
[7] Johan De Vriendt, Philippe Lainé, Christophe Lerouge, and Xiaofeng Xu, Alcatel, "Mobile Network Evolution: A Revolution on the Move", IEEE Communications Magazine, vol.40, no.4, Apr. 2002, pp. 104-111.
[8] Yaxin Cao and Victor O. K. Li, "Scheduling Algorithms in Broad-Band Wireless Networks", Proceedings of IEEE, vol.89, no.1, Jan. 2001, pp. 76-87.
[9] P. Bhagwat, A. Krishna, and S. Tripathi, "Enhancing throughput over wireless LAN's using channel state dependent packet scheduling," in Proc. INFOCOM'96, Mar. 1996, pp. 1133-1140.
[10] C. Fragouli, V. Sivaraman, and M. Srivastava, "Controlled multimedia wireless link sharing via enhanced class-based queuing with channel-state dependent packet scheduling," in Proc. INFOCOM'98, vol. 2, Mar. 1998, pp. 572-580.
[11] S. Lu and V. Bharghavan, "Fair scheduling in wireless packet networks," IEEE/ACM Trans. Networking, vol. 7, no. 4, 1999, pp. 473-489.
[12] L. Kleinrock, Queueing Systems, vol. 2: Computer Applications. New York: Wiley, 1976.
[13] A. Demers, S. Keshav, and S. Shenker, "Analysis and simulation of a fair queueing algorithm," in Proc. ACM SIGCOMM'89, 1989, pp. 3-12.
[14] T. S. Eugene Ng, I. Stoica, and H. Zhang, "Packet fair queueing algorithms for wireless networks with location-dependent errors," in Proc. INFOCOM98, Mar. 1998, pp. 1103-1111.
[15] P. Ramanathan and P. Agrawal, "Adapting packet fair queueing algorithms to wireless networks," in ACM/IEEE MOBICOM'98, Dallas, TX, pp. 1-9.
[16] J. Gomez, A. T. Campbell, and H. Morikawa, "The Havana framework for supporting application and channel dependent QoS in wireless networks," in Proc. ICNP'99, Nov. 1999, pp. 235-244.
[17] M. Shreedhar and G. Varghese, "Efficient fair queueing using deficit round robin," in Proc. ACMSIGCOMM'95, Berkeley, CA, 1995, pp. 231-242.
[18] Paul Bender et al, "CDMA/HDR: A Bandwidth-Efficient High-Speed Wireless Data Service for Nomadic Users", IEEE Communication Magazine, vol. 38, no.6, July 2000, pp. 70-77.
[19] A. Jalali, R. Padovani and R. Pankaj, "Data Throughput of CDMA-HDR a High Efficiency-High Data Rate Personal Communication Wireless Systems", IEEE VTC, 2000.
[20] F. Kelly, "Charging and rate control for elastic traffic," European Transactions on Telecommunications, vol. 8, pp. 33-37, 1997.
[21] Sanjay Shakkottai and Alexander Stolyar, "Scheduling for Multiple flows Sharing a Time-Varying Channel: The Exponential Rule", Translations of the AMS, A volume in memory of F. Karpelevich, Providence, R.I.: American Mathematical Society, 2001.
[22] Randy H. Katz, "Adaptation and Mobility in Wireless Information Systems", IEEE Personal Communications Magazine, vol.1, no.1, First Quarter，1994.
[23] Theodore S. Rappaport, "Wireless Communications Principles and Practice", Prentice Hall Inc., 1996, pp. 139-298. (中译本：蔡涛, 李旭, 杜振民. "无线通信原理与应用", 北京, 电子工业出版社, 1999, pp. 50-145)
[24] M. Gudmundson, "Correlation model for shadow fading in mobile radio systems", IEE Electronics Letters, vol. 27, Nov 1991, pp. 2145-2146.
[25] R. H. Clarke, "A Statistical Theory of Mobile-radio Reception", Bell Systems Technical Journal, Vol. 47, 1968, pp. 957-1000.
[26] B.D. Fritchman, "A Binary Channel Characterization Using Partitioned Markov Chains", IEEE Transactions on Information Theory, Vol. 13, No. 2, Apr. 1967, pp. 221-227.
[27] S. Tsai, "Markov Characterization of the HF Channel", IEEE Transactions on Communication Technology, Vol. 17, No. 1, Feb. 1969, pp. 24-32.
[28] H.S. Wang, and N. Moayeri, "Finite-State Markov Channel-A Useful Model for Radio Communication Channels", IEEE Transactions on Vehicular Technology, Vol. 44, No. 1, Nov. 1995, pp. 163-171.
[29] M. Zorzi, R.R. Rao, and L.B. Milstein, "On the Accuracy of a First-Order Markov Model for Data Transmission on Fading Channels", Proc. IEEE ICUPC'95, Nov. 1995, pp. 211-215.
[30] H. Wang and N. Moayeri, "Finite state Markov channel-A useful model for radio communication channels," IEEE Trans. Veh. Technol., Feb. 1995, pp. 163-171.
[31] Charles Chien et al, "Adaptive Radio for Multimedia Wireless Links", IEEE Journal on Selected Areas in Communications, vol.17, no.5, May 1999, pp. 793-813.
[32] ISO/IEC 14496-1, "Information Technology-Coding of Audio-visual Objects, Part 1: Systems", ISO/IEC JT1/SC 29/WG 11 Draft International Standard, Dec. 1998.
[33] 3GPP, "Universal Mobile Telecommunication System (UMTS); QoS Concept and Architecture," TS 23.107 v.3.5.0, http://www.3gpp.org, 1999.
[34] Mohamed N. Moustafa, Ibrahim Habib, Mahmoud Naghshineh and Mohsen Guizani, "QoS-Enabled Broadband Mobile Access to Wireline Networks", IEEE Communications Maganize, vol. 40, no. 4, Apr 2002, pp. 50-56.
[35] D. I. Birx, "Chaotic oscillator and CMFFNSfor signal detection in noise environments", IEEE International Joint Conference on Neural Networks, 1992, 22: 821-888.
[36] 王冠宇，陶国良，陈行，等. 混沌振子在强噪声背景信号检测中的应用, 仪器仪表学报, 1997, 18(2): 209-212.
[37] 王冠宇，陈大军，林建亚，等. Duffing 振子微弱信号检测方法的统计特性研究. 电子学报. 1998，26(10): 38-43.
[38] Xun-he Yin, Ru-peng Feng, Yong Ren, "A kind of new amplifier". Chinese Physics, 2000, 9(2): 97-99.
[39] 童勤业，严筱刚，孔军，薛宗琪. “混沌”理论在测量中的应用. 电子科学学刊，1999，21(1): 42-49.
[40] Suqin Chen, Qinye Tong, Ruiyue Yuan. "N-nary A/D conversion with nonlinear dynamics method". Proceedings of the 1999 3 rd International Symposium on Test and Measurement. Xi'an, China: Chinese Society of Modern Technical Equipments: 1999, 193-194.
[41] 陈素琴，袁瑞跃，童勤业. 多进制式混沌A/D变换器研究. 计量学报，2000， 21(1): 12-16.
[42] A. Yilmaz, K.C. Wong, K.S. Chao. "A switched-current cyclic A/D conversion technique". Proceedings of the 37th Midwest Symposium on Circuits and Systems, IEEE, 1994. v2: 1155-1158.
[43] Mikael Gustavsson, J. Jacob Wikner, Nianxiong Nick Tan. CMOS Data Converters for communications, Boston: Kluwer Academic Publishers, 2000.