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Efficient strategies for representing and evaluating the effect of propagation impairments on the performance… Liu, Xia 2005

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EFFICIENT STRATEGIES FOR REPRESENTING AND EVALUATING THE EFFECT OF PROPAGATION IMPAIRMENTS ON THE PERFORMANCE OF WIRELESS COMMUNICATIONS SYSTEMS by X I A L I U B . S c , The U n i v e r s i t y o f C h a n g C h u n Post and Te l ecommunica t i on , 1996 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S E lec t r i ca l and Computer Eng inee r ing T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l 2005 © X i a L i u , 2005 A b s t r a c t Current methods for assessing the effect of propagation impairments on the link-level performance of wireless communications systems either yield results that apply to certain specific channel conditions or only produce statistical summaries of system performance over a wider range of conditions. Here, we show that a polynomial response surface model can efficiently represent the bit error rate performance of a wireless communications system as a function of standard channel parameters. The result may be used as an complete yet compact equipment performance model useful in higher level system simulation or as an aid to visualization of system performance. When two channel parameters are considered, a fifth-order polynomial is sufficient to model the bit error rate performance over a reasonable range of channel conditions. This implies that a complete description of system performance can be captured using just twenty-one parameters. Furthermore, compared to regular sampling, adaptive sampling of the response surface can significantly reduce the time and effort required to generate a response surface model from physical layer simulations. While the results presented here were generated by simulation, our response surface model and adaptive sampling method could also be applied to experimental methods that assess the link-level performance of wireless communications systems using RF channel emulators. ii Table of Contents Abstract ii Table of Contents iii List of Tables vi List of Figures vii Acknowledgments ix Chapter 1 Introduction 1 Chapter 2 Approaches to Characterization of the Performance of Wireless Communications Systems in the Presence of Propagation Impairments 4 2.1 Introduction 4 2.2 The System Signature Concept 5 2.3 Current Approaches 9 2.4 Some New Approaches 11 2.4.1 Propagation Performance Envelope 11 2.4.2 Performance Response Surface Modeling 13 2.4.3 Adaptive Sampling 15 Chapter 3 Use of Response Surfaces to Characterize the Performance of Wireless Communications System in the Presence of Propagation Impairments 17 3.1 Introduction 17 3.2 Approaches to Visualizing the Performance of Wireless Communications Systems in the Presence of Propagation Impairments 18 3.2.1 Two-Dimensional Representation of the Performance of Wireless Communications Systems 18 iii 3.2.2 Three-Dimensional Representation of the Performance of Wireless Communications Systems 20 3.2.3 Propagation Performance Envelope 20 3.3 Response Surface Modeling 24 3.3.1 The Concept of the Response Surface Model 25 3.3.2 Procedure for Generating a Response Surface Model 25 3.3.2.1 Selection of an Appropriate Surface Model 25 3.3.2.2 Tools Used to Generate Response Surface Models '. 26 3.3.3 Verification of the Accuracy of Response Surface Models 27 3.3.3.1 Verification Method 27 3.3.3.2 First Example of Response Surface Modeling 28 3.3.3.3 A Second Example of Response Surface Models 31 3.4 Implementation of Response Surface Models in Different Applications 34 3.4.1 Application of Response Surface Models in Visualization of the Performance of Wireless Communication System 35 3.4.2 Application of Response Surface Models in Network Simulation 36 3.4.3 Application of Response Surface Models in Evaluation of the Performance of Equipment 37 3.4.4 Application of Response Surface Models in Comparing the Performance of Different Equipment 39 3.5 Conclusion 40 Chapter 4 Use of Adaptive Sampling to Simplify Determination of the Performance of Wireless Communications Systems in the Presence of Propagation Impairments 42 4.1 Introduction 42 iv 4.2 Concept 44 4.3 Implementation 47 4.4 Performance 50 4.5 Conclusion 61 Chapter 5 Conclusions and Recommendations 62 5.1 Conclusions 62 5.2 Recommendations for Further Work 63 References 65 v List of Tables Table 2-1 JTC channel model parameters for indoor office areas 9 Table 3-1 Comparison of RMSE obtained from different response surface models (the polynomial models are obtained by fitting the surface of B E R vs. RMS delay IT vs. Doppler spread with DBPSK modulation) 30 Table 3-2 Comparison of R M S E obtained from different response surface models (the polynomial models are obtained by fitting the surface of BER vs. RMS delay IT vs. Es/No with M S K modulation) 33 Table 4-1 RMSE comparison of adaptive and fixed rate sampling when they are tested by a simple function 52 Table 4-2 Total data points' comparison of adaptive and fixed rate sampling when tested by function 54 Table 4-3 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate BER vs. K vs. RMS delay/T for DBPSK modulation 57 Table 4-4 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate BER vs. Doppler spread vs. RMS delay/T for DQPSK modulation 59 vi L i s t o f F i g u r e s Fig 2-1 Performance comparison different modulation schemes with A W G N 4 Fig 2-2 M-curve for 8-PSK modem and 6.3 ns delay 6 Fig 2-3 PPE for a simple channel model with DBPSK modulation scheme 13 Fig 3-1 Two dimensional representation of the performance of wireless communication system in the presence of A W G N and delay spread 19 Fig 3-2 Three dimensional representation of the performance of wireless communication system 20 Fig 3-3 Flow chart of finding satisfied parameter sets 22 Fig 3-4 PPE for a simple wireless communication system 23 Fig 3-5 Simulation model of wireless communication system for the generation of PPE 24 Fig 3-6 Comparison of B E R surfaces (BER vs. RMS delay/T vs. Doppler spread for DBPSK modulation scheme) generated by simulation and response surface models 29 Fig 3-7 Distribution of relative errors obtained from different polynomial models (the polynomial models are obtained by fitting the surface of B E R vs. RMS delay /T vs. Doppler spread with DBPSK modulation) 31 Fig 3-8 Comparison of BER surfaces (BER vs. RMS delay/T vs. Es/No for M S K modulation scheme) generated by simulation and response surface models 33 Fig 3-9 Distribution of relative errors obtained from different polynomial models (the polynomial models are obtained by fitting the surface of BER vs. RMS delay IT vs. Es/No with M S K modulation) 34 Fig 3-10 Performance of wireless communication system (BER vs. RMS delay/T vs. Es/No for M S K modulation scheme) show in nlintool GUI 36 vii Fig 3-11 Wireless performance prediction using equipment performance models (response surface model) 37 Fig 3-12 A test bed for conducting RF performance testing 38 Fig 4-1 Use of adaptive methods to concentrate samples in regions with complicated profiles. 43 Fig 4-2 The criterion to decide another division 45 Fig 4-3 Flow chart of Adaptive Sampling 49 Fig 4-4 Comparison of adaptive and Fixed rate sampling when they are tested by function 52 Fig 4-5 RMSE comparison of adaptive and fixed rate sampling when they are tested by a function 53 Fig 4-6 Total data points comparison of adaptive and fixed rate sampling when tested by a simple function 55 Fig 4-7 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate B E R vs. K vs. RMS delay/T for DBPSK modulation 56 Fig 4-8 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate B E R vs. Doppler spread vs. RMS delay/T for DQPSK modulation 58 Fig 4-9 Simulation time comparison of adaptive and fixed rate sampling when the sampling algorithms are tested in simulation 60 Fig 4-10 R M S E comparison of adaptive and fixed rate sampling when the sampling algorithms are tested in simulation 60 viii Acknowledgments I would like to take this opportunity to thank the people who played an important role in this work. At first, I would like to thank my supervisor, Prof. Dave Michelson, for his continuous advice and guidance for my thesis project. Prof. Michelson also provided me many chances to contact with industry people who are interested in my project, so we can make my thesis project more practical. It's a great pleasure to thank all the group members of the Radio Science Lab of U B C for their friendship and technical suggestions. Under the organization of Prof. Dave Michelson, I have chance to present and discuss my work to other group members at our group meeting. The group members, especially, Steve Ma, Chengyu Wang and Chris Hynes gave me much good advice. Chris Hynes provided the initial TCL script for the generation of Propagation Performance Envelope (PPE). I would also like to thank all the friends that I made in Vancouver, especially Peter Sun, Carrier Wang, XiaoXiao, Leo L i , Rachel Wu, Chris Lin, XiaoLei Wang, Rong L i , Bruce Wang and LiMei Zhang. Special thanks give to my husband, David Yang, for his love, patience and support in my research and life. Finally, I would like to specially acknowledge my parents, ShouXian Liu and ZhuYe Sheng, for their love and support. I also want to thank my sister, Jeanette Liu, and her husband, Tim Yu, for their support. ix C h a p t e r 1 I n t r o d u c t i o n This thesis is concerned w i t h the development o f efficient strategies for representing and evaluating the effect o f propagation impairments on the l i n k - l e v e l performance o f wireless communica t ions systems. In very s imple cases, the l i nk - l eve l performance o f a wireless communica t ions system can be adequately characterized s i m p l y by determining the manner i n w h i c h bi t error rate ( B E R ) degrades as the signal-to-noise ratio ( S N R ) at the receiver input decreases. Such a result provides a complete descr ipt ion o f l i nk - l eve l performance, but on ly under relat ively benign condit ions, i.e., a t ime-invariant and frequency-flat channel . H o w e v e r , as wireless systems move to higher frequencies, become more complex , or are dep loyed i n more chal lenging environments, it is often necessary to account for the effects o f a t ime-vary ing and/or frequency selective channel . The s imple B E R curve becomes a complex funct ion o f many variables inc lud ing S N R at the receiver input, delay spread, Dopp le r spread, and R i c i a n K-fac tor . Current methods for assessing the effect o f propagat ion impairments on l i nk - l eve l performance either y i e l d results that apply on ly to certain specif ic channel condit ions or on ly produce statistical summaries o f l i nk - l eve l performance over a part icular set o f channel condit ions. W h i l e adequate for some purposes, a more complete descript ion o f l ink - l eve l performance is required for applications such as system-level s imula t ion or v isua l iza t ion o f l i nk -level performance. In p r inc ip le , one cou ld provide such a descr ip t ion s i m p l y by comput ing or measuring the bi t error rate or s imi la r metric for a l l possible combinat ions o f the mul t ip le 1 parameters that describe the propagation channel. However, this approach is both cumbersome and time-consuming. In this thesis, we show how the complex manner in which propagation impairments jointly affect link-level performance may be efficiently yet completely captured in the form of a polynomial response surface model. While our results are generated from simulations, they could equally as well be generated from measurements of actual equipment using an RF channel emulator and a bit error rate test set. The results may be used as an efficient lookup table (or equipment response model) useful in system-level simulations or as a compact representation useful as an aid to visualization of system performance over a broad range of channel conditions. Moreover, we show that the time and effort required to evaluate the model can be significantly reduced through adaptive sampling of the propagation parameter space. Our results have considerable practical significance. For example, early results of our work have been presented to and have influenced the outcome of the IEEE 802.1 IT task group on wireless performance prediction. This thesis is organized as follows: In Chapter Two, we review the strengths and limitations of alternative approaches for evaluating, representing, and comparing the performance of wireless communications systems in the presence of propagation impairments and suggest ways in which these schemes can be enhanced or improved. In Chapter Three, we show that polynomial response surface models can be used to efficiently represent the performance of wireless communications systems as a function of propagation channel parameters for the purposes of both visualization of link-level performance and higher-level system simulation. 2 In Chapter Four, we show that adaptive sampling can be used to significantly reduce the effort required to generate a polynomial surface response model from physical layer simulations. In Chapter Five, we draw conclusions, assess the limitations of the present work, and offer recommendations for future work. 3 C h a p t e r 2 A p p r o a c h e s t o C h a r a c t e r i z a t i o n o f t h e P e r f o r m a n c e o f W i r e l e s s C o m m u n i c a t i o n s S y s t e m s i n t h e P r e s e n c e o f P r o p a g a t i o n I m p a i r m e n t s 2.1 Introduction In very s imple cases, the l i n k - l e v e l performance o f a wireless communica t ions system can be adequately characterized s i m p l y by determining the manner i n w h i c h bit error rate ( B E R ) degrades as the signal-to-noise ratio ( S N R ) at the receiver input decreases, as shown in F i g . 2.1. Such a result provides a complete descript ion o f l i n k - l e v e l performance, but only under relat ively benign condi t ions , i.e., a t ime-invariant and frequency-flat channel . 10 E^t error rate vs Eb /No for two and four phase psk with A W G N o -i—< CCS 10 10 •&— B P S K - * — D Q P S K .: * D B P S K -o 10 -4 -5 10 -6 0 2 4 6 8 Eb/No (dB) 14 F i g 2-1 Performance compar ison different modula t ion schemes w i t h A W G N 4 As wireless systems move to higher frequencies, become more complex, or are deployed in more challenging environments, it is often necessary to account for the effects of a time-varying and/or frequency selective channel. The need to characterize their performance and verify their reliability over a broad range of channel conditions is becoming increasingly important. Theorists, systems engineers, hardware designers, and systems integrators all have an interest in the results. In this chapter, we review alternative approaches to characterizing the performance of wireless communications systems in the presence of propagation impairments. In Section 2.2, we review the system signature concept introduced by Emshwiller and others in the 1970's for characterizing the performance of long haul digital microwave radios in the presence of multipath fading. In Section 2.3, we review current approaches to the problem of characterizing the performance of wireless communications systems in the presence of propagation impairments. In Section 2.4, we introduce possible ways to address the shortcomings of current approaches, some of which will be developed further in Chapters Three and Four. 2.2 The System Signature Concept Among the first efforts to characterize the effect of a time-dispersive (frequency selective) channel on the performance of a digital wireless communications system was the system signature concept. Introduced by Emshwiller [1] in 1978, it was intended as a method for capturing the manner in which link outage time depends on multipath delay. An outage event is said to occur when the system performance (BER) degrades to some threshold for not more than ten consecutive seconds. If the event lasts for more than ten consecutive seconds, the link is simply unavailable. Emshwiller chose a BER of 10"3 as a performance threshold and used a simple two-ray fading model to estimate the outage time. In this channel model, three parameters are considered. One of them is delay x of the interfering ray, which is fixed to some value. The other two parameters are amplitude A. of the second ray and the frequency offset from the carrier. A l l of these parameters have reasonable physical meanings. Through computer simulations of an 8PSK modem, he obtained a set of curves corresponding to amplitude X of delayed path versus frequency offset. One curve which is obtained when x equal to 6.3 ns is shown in Fig 2-2. In reference to their shapes, this set of curves is so called M-curves. The two dimensional contours (M-curves) separate the regions where the system performance is acceptable from those regions where system performance is unacceptable. x = 6.3 ns; 8-PSK with roll-off = 0.5 B E R threshold = 10' 3 -6 -2 0 2 6 Frequency offset, MHz Fig 2-2 M-curve for 8-PSK modem and 6.3 ns delay In 1978, Greenstein and Prabhu [2] and Jakes [3] suggested a similar idea at the same conference. The extended versions of their papers [4-5] were published in 1979 in IEEE Transactions on Communications. They both used two-path multipath model to predict the system outage. They assumed that the direct path has fixed amplitude, the amplitude of the delayed path follows an exponential distribution, and the delay for the second path is also exponentially distributed. Different criteria are used to decide the existence of outage event. BER is chosen as performance threshold by Greenstein and Prabhu; but the BER value (10"6) is much smaller than the one chosen by Emshwiller. Jakes said that an outage event occurs when the peak-to-peak delay within the signal band exceeds some constant times the symbol period. A l l the works mentioned above assume that without an equalizer, outage time was dominated by multipath frequency-selective fading rather than by Additive White Gaussian Noise. Accordingly, A W G N was not considered in the channel model. In 1978, Rummler and Lundgren developed a simplified three-path model [6]-[7]. Considering thermal noise, four parameters should be included into the channel model; where parameter a is the amplitude of the direct path; parameter b is the relative amplitude of the delayed path; r is still the delay of the second path which is chosen as 6.3 ns and fo is the notch frequency. The amplitude of all the paths are scaled as A = -20 log a; and B = -20 log (1-b). For the radio system they studied, Limdgren and Rummler also found that outage time is dominated by fading rather than by thermal noise. Accordingly, they removed the parameter A from the model, and obtained a set of curves which is corresponded to parameter B and fo. These curves are called W-curves in reference to their shapes. Rummler later implemented this three-path model in an effort to predict multipath outage in the presence of thermal noise [8]. In this case, M-curves are no longer adequate to represent the system signature. A set of curves (critical A - B curves) are used to replace W-curves. In the critical A-B curves, sets of fade level A and relative notch depth B which produces a 10" BER at some specific notch frequency are plotted on a two dimensional graph. Besides the two-path 7 channel models and three-path channel models, polynomial channel models were also investigated by several researchers [9-11]. To predict system performance, an accurate channel model is very important. The channel models introduced above are all statistical models. They are only applicable to the line-of-sight case. For simplicity, the channel is assumed to include at most three paths. Even for three-path models, the first path usually has little difference with the second path, and the delay between the two paths can always be neglected. These assumptions may not hold in other environments, however, and must be used with caution. It was concluded by Greenstein and Shafi [12] that there were three approaches used to find the parameter region, over which the performance threshold is exceeded. They are laboratory measurements, mathematical analysis and computer simulation. At that time, each approach had its limitations and advantages. The simplicity of the channel model makes the mathematical analysis possible and easy to be done. But the shortage of powerful and convenient measurement instruments at the time made laboratory measurements difficult; and the slow speed of computers at the time made computer simulation problematic. The problems encountered at that time, including selection of an appropriate channel model or obtaining the system signature, can be solved today. Currently, many multipath fading channel models are developed for various environments, such as the Rayleigh fading channel model and the Rician fading channel model, etc. These models are obviously more complex than the statistical models used in 1980s, so the mathematical analysis for the system performance is much more complicated. But the powerful and convenient measurement instruments (such as various channel emulators) make it much easier to conduct laboratory measurements. Computer simulation is another good choice given the existence of powerful physical layer simulation software and high-speed computers. 8 2.3 Current Approaches Two common methods are currently used to assess the link-layer performance of wireless communications systems over time-varying or frequency-selective channels. In the first method, the channel is modeled as a finite impulse response (FIR) filter with specified tap delays and amplitudes that represent either typical or bad conditions in various environments. Such models are typically used to assess the ability of some signaling scheme (or some equipment) to deliver a specified level of performance (e.g., a BER greater than some minimum value) in a particular class of environment. Table 2-1 JTC channel model parameters for indoor office areas Channel A Channel B Channel C TAP Relative Average Relative Average Relative Average Doppler Delay Power Delay Power Delay Power spectrum (nsec) (dB) (nsec) (dB) (nsec) (dB) 1 0 0 0 0 0 0 Flat 2 50 -3.6 50 -1.6 100 -0.9 Flat 3 100 -7.2 150 -4.7 150 -1.4 Flat 4 325 -10.1 500 -2.6 Flat 5 550 -17.1 550 -5 Flat 6 700 -21.7 1,125 -1.2 Flat 7 1,650 -10 Flat 8 2,375 -21.7 Flat Many international or national organizations that contribute to standardization of communication systems, such as the Institute of Electrical and Electronics Engineers (IEEE) and the European Telecommunications Standards Institute (ETSI) provide some standard models for modeling and simulating mobile radio channels in various environments. For example, Table 2-1 shows the channel model parameters recommended by JTC (Joint Technical Committee) for PCS Air Interface Standards. This standard is developed for indoor office areas. 9 F r o m Tab le 2-1 w e can see that a l l the parameters are f ixed to some specific values i n order to represent a typ ica l condi t ion encountered i n indoor office areas. Thus , the performance o f the system we obtain through this standard mode l is also specif ic for this condi t ion. There is a lways some performance threshold (e.g., B E R ) used to determine i f the system performance is acceptable. In many cases, a s imple pass or fai l test under standard environments forms the basis o f contractual obl igat ions between equipment vendors and system operators. W h e n comparison o f the performance o f alternative schemes or equipments is considered, the s imple pass or fai l test is not enough. The compar i son should be made i n a broad variety o f environments i n order to determine the l imi t s o f performance, not on ly i n standard condi t ions . In the second method, the dynamic nature o f the propagat ion environment is considered. A complete first-order statistical descript ion o f the F I R mode l parameters applicable to a particular environment is g iven and used to generate a range o f channel behavior w h i c h is typ ica l o f the environment . Th i s method can also be ca l l ed D y n a m i c Env i ronment Emula t i on ( D E E ) . The D E E a l lows the user to cascade mul t ip le static parameter "states" to emulate a dynamic propagat ion environment. In D E E , a group o f parameters w h i c h can be defined by the first method is represented as a state. A sequence o f such states creates a test scenario. W i t h this method, the range o f system performance and the probabi l i ty o f system outage i n specific environments can be predicted. Th i s method can simulate the system i n a real environment i f the complete first-order statistical descr ipt ion o f the F I R mode l parameters for that part icular environment is available. However , the detailed propagation models o f specific environments are not easy to obtain. E v e n i f the detailed propagation models can be obtained, the prof i le generations for a l l the states are s t i l l an important issue. W h e n this method is appl ied to network l eve l s imulat ions, the long 10 simulation time for DEE is also unacceptable. For the comparison of two schemes or pieces of equipment, this method is obviously better than the first one. Alternative equipment or schemes can be compared in dynamic environments. The system outage time or performance range can be compared. But these comparisons do not provide much physical insight into the reasons for the outages. 2.4 Some New Approaches After reviewing current approaches to the characterization of the performance of wireless communications systems in the presence of propagation impairments, we know that these schemes suffer from significant limitations when applied to network level simulations or comparison of the performance of alternative schemes when detailed propagation models of specific environments are not available. In this section, some new concepts will be introduced that seek to improve the current approaches. 2.4.1 Propagation Performance Envelope The performance of wireless communications systems is usually represented by a set of curves where the BER (Bit Error Rate) is plotted against one of the channel parameters. Usually the BER is plotted against Es/No (the energy per symbol over the noise spectral density) when the A W G N (Additive White Gaussian Noise) is assumed to be the only impairment during propagation. When multipath fading is considered, the BER may be plotted against some other channel parameters, such as Rician K factor, RMS delay or Doppler spread, etc., as has been done by various researchers [13-16]. In these papers, the performance is represented in a two dimensional graph. In these graphs, the BER was plotted against one of the channel parameters with another channel parameter changed by step. For example, a set of curves are obtained when the BER is plotted against RMS delay/symbol period (t/T) with different Es/No. 11 This representation can be shown in another more intui t ive and direct way . Because performance is related to two channel parameters, the B E R can be plot ted i n a three dimensional graph. A c t u a l l y , w e want to consider more channel parameters s imul taneously when assessing the performance o f wireless communicat ions systems. The system signature concept suggests a possible approach. The performance o f the wireless communica t ions systems can be represented by system signature i n a two d imensional graph, and at the same t ime, two parameters are considered s imultaneously. F o r the sake o f v isua l iza t ion , at most three-dimensional graphs can be considered to represent the B E R . W h e n the idea o f the system signature is implemented i n three dimensions, a new concept is introduced, the propagation performance envelope ( P P E ) . O b v i o u s l y , the channel mode l used to generate P P E is different f rom the statistical models used to generate system signature. Standard channel models that can present various propagation environments are used to generate the system performance i n those environments . S i m i l a r to the system signature, the propagation performance envelope is s t i l l the boundary between the parts where the performance is unacceptable and the parts where the performance is acceptable. The propagation performance envelope is a surface defined by a set o f channel parameters for w h i c h a certain performance cr i ter ion (usual ly the B E R ) is satisfied. The P P E can be obtained by three steps. 1) A performance threshold is decided (Usua l ly the B E R ) . 2) W h e n two o f the channel parameters are changed by step, the third channel parameter is found by binary search to meet the B E R threshold. 3) The set o f parameters w h i c h satisfied the B E R threshold are plotted i n a three d imens iona l graph. 12 An example of PPE is shown in Fig 2-3 to compare with the system signature shown in Fig 2-2.When the performance of wireless communications systems is represented by PPE, the effects of three channel parameters on the system performance can be shown in one graph. 0.2. Maximum Doppler Freq(Hz) 0 10 EsNo (dB) Fig 2-3 PPE for a simple channel model with DBPSK modulation scheme 2.4.2 Performance Response Surface Modeling For the purposes of visualization, representing the system performance by some surfaces as three-dimensional graphs is convenient, intuitive and complete. Different from the current approaches to characterizing the performance of wireless communications systems, the system performance is shown not only for some specific environments, but for any environment. Each set of parameters can represent one specific condition. Even i f the detailed propagation models 13 of specific environments are not available, the comparison of the performance of alternative schemes can also be done successfully. But, the implementation of this representation of performance into the network level simulation is obviously inefficient. To make the representation more compact, a new concept performance response surface modeling is introduced. Many of the notions associated with response surface modeling come from the Design of Experiments (DOE) [17]. The response surface models are used in DOE to simplify the results generated from simulations or experiments. Here, the response surface model can be used to efficiently represent the performance of wireless communications systems as a function of propagation channel parameters. In Matlab, there is a toolbox that is designed specifically for response surface modeling. It makes the modeling of performance surfaces of wireless communications systems much easier. Two kinds of models are used to generate response surface models in this project. One is the quadratic polynomial model (the order of the polynomial is two), and the other is the higher-order polynomial model (the order of the polynomial is bigger than two). In the quadratic model, the response surface is modeled as a second-order polynomial: 2 2 y = bo + bixi + b2X2 + byz\X2 + bjXi + b&2 (2-1) For the higher-order polynomials, as the polynomial order increases, so does the accuracy of the model. But at the same time, the model becomes more complicated and more parameters are needed to specify the response surface. For a third-order polynomial, 10 parameters are needed; for fourth-order polynomial, 15 parameters are needed; and for fifth-order polynomial, 21 parameters are needed. 14 Some transformation of outputs are always used (e.g., log, exp, square, square root, inverse) to aid model fitting and optimization in DOE. In the case of performance surface modeling of wireless communications systems, BER surfaces are modeled, so a log transformation of the BER is often done in order to ease the modeling. To verify the accuracy of the response surface model, two methods are used to calculate the error. In the first method, the root mean square value is given for the residue (error between estimated value using generated model and the measured value). In the case of performance surface modeling of wireless communications systems, log format of B E R is used. When the BER is high (e.g., 10"1), the small error (e.g., 0.0001) is not a big problem. But when the BER is small (e.g., 10"4), the same small error (0.0001) is really a big problem. So the root mean square error which is calculated by absolute errors does not have so much meaning to measure the error. In the second method, the relative error is calculated as: , error between estimated value and measured value relative error = (2-2) measured value Then the root mean square of the relative error is calculated. 2.4.3 Adaptive Sampling To obtain an accurate response surface model, the number of points that make up the surfaces must be large enough. However, a large number of points implies a long simulation or measurement time. To reduce the effort required to generate a surface response model for physical layer simulation or emulation, we introduce an adaptive sampling algorithm. The concept of adaptive sampling is that fewer samples are needed with variable sampling rate while maintaining the similar response characteristics compared with traditional fixed rate 15 simulation. Considering the simulation or experimental time, the adaptive sampling algorithm developed for response surface model has to be simple and easy to implement. Adaptive sampling can be implemented by two steps. In step one, a fixed-rate coarse sampling is done to generate initial response surface. In step two, each unit that makes up of the whole surface is divided into sub-units if some criterions are unsatisfied. The criterion used here is the angle between the original unit and sub-units. The angle has to be compared with some threshold to decide if the unit is small enough. There is another threshold the maximum number of divisions which is used to avoid infinite division when response surface changes suddenly and dramatically. This step is done repeatedly until all the units and sub-units satisfy criterion or the maximum number of divisions is met. Response surface modeling and adaptive sampling are considered in further detail in Chapters Three and Four respectively. 16 C h a p t e r 3 U s e o f R e s p o n s e S u r f a c e s t o C h a r a c t e r i z e t h e P e r f o r m a n c e o f W i r e l e s s C o m m u n i c a t i o n s S y s t e m i n t h e P r e s e n c e o f P r o p a g a t i o n I m p a i r m e n t s 3.1 Introduction A t present, three c o m m o n methods are used to assess the l ink - l aye r performance o f wireless communica t ions systems. In the first method, A d d i t i v e W h i t e Gauss ian N o i s e ( A W G N ) is assumed to be the on ly propagation impairment and the B i t E r r o r Rate ( B E R ) is plotted against E s / N o (the energy per s y m b o l over the noise spectral density). In the second method, the channel is mode led as a F in i te Impulse Response ( F I R ) filter w i t h specif ied tap delays, amplitudes, and fading characteristics that represent either typ ica l or bad condit ions in various environments as appropriate. In the third method, a complete first-order statistical description o f the F I R mode l parameters appl icable to a particular environment are g iven and used to generate a range o f channel behavior w h i c h is typ ica l o f that environment. T h i s method is often ca l led dynamic environment emula t ion (or s imulat ion) . A l t h o u g h these approaches are adequate for their intended purposes, a more complete descript ion is required for applications such as system-level s imula t ion (using packages such as O P N E T or ns2) or v i sua l i za t ion o f l ink- layer performance across a broad range o f channel condit ions. In p r inc ip le , one c o u l d provide a complete descr ipt ion o f l i n k - l e v e l performance s imp ly by comput ing or measuring the bit error rate or s imi l a r metr ic for a l l possible combinat ions o f the mul t ip le parameters that describe the propagat ion channel . Howeve r , this approach is both cumbersome and t ime-consuming. In this chapter, we show h o w the complex manner i n w h i c h propagat ion impairments j o i n t l y affect l i n k - l e v e l performance may be 17 efficiently yet completely captured in the form of a polynomial response surface model. The result may be used as an efficient lookup table (or response surface model) useful in system-level simulations or as a compact representation useful as an aid to visualization of system performance over a broad range of channel conditions. The remainder of this chapter is organized as follows. In Section 3.2, we review methods used to visualize the performance of wireless communications systems in the presence of propagation impairments. In Section 3.3, the response surface model will be introduced to simplify the performance surfaces. In Section 3.4, potential applications of response surface models are considered. In Section 3.5, conclusions are drawn. 3.2 Approaches to Visualizing the Performance of Wireless Communications Systems in the Presence of Propagation Impairments 3.2.1 Two-Dimensional Representation of the Performance of Wireless Communications Systems At present, the performance of wireless communications systems is usually presented by a set of curves corresponding to BER and Es/No, when A W G N is assumed to be the only propagation impairment. When multipath fading is a significant propagation impairment, the performance of wireless communications systems can also be represented by curves corresponding to B E R and one of the multi-path fading channel parameters, such as the Rician K factor, RMS delay or Doppler spread. In the antecedent representation, only the relation between one channel parameter and BER can be considered. To include more channel parameters, a set of curves is usually plotted in a two-dimensional graph. In the graph, BER was plotted against one of the channel 18 parameters w i t h another channel parameter fixed at different values. T h i s representation has been used by many authors [ 1 3 , 14 , 15 , 16] . F o r example , B E R vs. R M S de lay /symbol per iod ( T / T ) VS. E S / N O is shown i n F i g 3 - 1 . 0.6 0.8 R M S d e l a y / T F i g 3-1 T w o d imens iona l representation o f the performance o f wire less communica t ion system i n the presence o f A W G N and delay spread. In this example , a s imple wireless communica t ion system w i t h D B P S K modula t ion and demodulat ion schemes is considered. B E R is plotted against R M S de lay /T w i t h E s / N o changing by step (5dB) . The parameter settings are g iven i n the legend o f F igure 3 - 1 . 1 9 3.2.2 Three-Dimensional Representation of the Performance of Wireless Communications Systems This relationship shown in the last example can also be represented in a three-dimensional graph, and is shown in Fig 3-2. -0.5 -1 -s -1.5 > en - 2 . L U CD -2.5. a i o - 3 . -3.5. -4> 1.5 RMSde lay / r Fig 3-2 Three dimensional representation of the performance of wireless communication system 3.2.3 Propagation Performance Envelope Either the two-dimensional or three-dimensional representation can present at most two channel parameters simultaneously. To include one more channel parameter, a new concept, the Propagation Performance Envelope (PPE), which is somewhat similar to the system signature, is 20 introduced. The propagat ion performance envelope is the boundary between the parts where the system performance is unacceptable and the parts where the system performance is acceptable. The propagation performance is a surface defined by a set o f channel parameters for w h i c h a certain performance cr i ter ion (usually is chosen as B E R ) is satisfied. The P P E can be obtained by three steps. 1) First, a performance threshold and performance threshold tolerance should be chosen. U s u a l l y the B E R is chosen as performance threshold. It is diff icul t to find a set o f parameters that meet the B E R threshold exact ly either b y s imula t ion or laboratory experiment. So , some B E R tolerance is chosen to ease the finding o f satisfied parameter sets. W h e n the B E R falls between B E R threshold - B E R tolerance and B E R threshold + B E R tolerance, the performance threshold is met. 2) Find the parameter sets which satisfy the performance threshold. F o r the visua l iza t ion , i n this chapter, the parameter set on ly includes three channel parameters. The whole process o f f ind ing such parameter sets is shown i n F i g 3-3. In the f low chart, the three channel parameters are presented as A , B and C . The m i n i m u m and m a x i m u m values o f the three channel parameters are chosen first. In the process o f f inding satisfied parameter sets, the values o f first two channel parameters A and B are changed by two loops and the th i rd channel parameter C is found by b inary search to meet the B E R threshold. Before the start o f binary search, a dec is ion is made to save t ime i f there is no need to conduct b inary search. That is when the B E R w i t h the best case o f parameter C is s t i l l worse than the B E R threshold, or the B E R w i t h the worst case o f parameter C is s t i l l better than the B E R threshold. Af te r a l l the parameter sets are found, the results are writ ten i n a text file for future analysis. 21 Start 1 Parameters initialization Loop for parameter A Loop for parameter B Binary search to find the parameter C so the BER threshold is satisfied Record the parameter set (A, B and C) which satisfied the BER threshold End of loop for parameter B End of loop for parameter A End N F i g 3-3 F l o w chart o f f inding satisfied parameter sets F o r the b inary search, the parameter C is chosen as the median value between the m a x i m u m and m i n i m u m values first. I f the B E R obtained f rom s imula t ion or emulat ion is bigger than threshold, change the parameter C toward the best case. F o r example, i f the parameter C is E s / N o or K factor, m a x i m u m value is the best case; i f the parameter C is R M S delay or D o p p l e r spread, m i n i m u m value is the best case. The changing step is the median value between the current value and the value o f the best case or worst case. A s imi la r process is f o l l owed when the obtained B E R is less than the threshold. In some cases, the l imitat ions o f s imula t ion or laboratory experiment may make it difficult or imposs ible to f ind some satisfied parameter sets. T o a v o i d an infinite binary search, another threshold, the m a x i m u m mumber o f b inary search, is introduced. 3) Plot the parameter sets which satisfy the BER threshold in a three dimensional graph. A n example o f P P E obtained f rom S i m u l i n k s imulat ion is shown i n F i g 3-4. F i g 3-4 P P E for a s imple wireless communica t ion system 23 The model of the wireless communication system expressed in Simulink is shown in Fig. 3-5. In this example, the wireless communication system is composed of a Random Binary Source, DBPSK modulation and demodulation blocks and channel. The channel includes Rician fading, Rayleigh fading and A W G N . In the Rayleigh fading block, the number of paths is chosen as five, and the Gain vector of the five paths is fixed as [0 -3 -6 -9 -12] dB. The BER threshold is chosen as 0.01, the tolerance is chosen as 2% of the BER threshold, and the maximum number of binary search is chosen as 10. Bernoulli Binary Bernoulli Random Binary Generator DBPSK DBPSK Modulator Baseband Tx Rx Error Rate Calculation Rician Fading Rician Fading Channel F — 2 Multipath Rayleigh Fading Multipath Rayleigh Fading Channel Error Rate Calculation AWGN AWGN Channel DBPSK Demodulator Baseband wviAh_r-DBPSK Fig 3-5 Simulation model of wireless communication system for the generation of PPE 3.3 Response Surface Modeling Although the three-dimensional representation of system performance is intuitive, the surface is represented by a great deal of data. This makes it awkward to use the raw data in network simulation or used for evaluation and comparison of system components. To simplify the use of performance surfaces, the response surface model is introduced in this chapter. 24 3.3.1 The Concept of the Response Surface Model The concept of the response surface model comes from the Design of Experiment (DOE) [17]. In DOE, the response surface model is designed to estimate interaction and even quadratic effects, and give an idea of the shape of response surface being investigated. For this reason, it is called the response surface model (RSM). The original objective of the R S M is to find optimal process settings, troubleshoot process problems and weak points, and make a process more robust against external and non-controllable influences. In DOE, the response surface is usually modeled as a mathematical function of a few continuous factors and the model parameters are estimated by regression design. In this chapter, response surface modeling is used to efficiently represent the performance of wireless communications systems as a function of propagation channel parameters. In this case, the response surfaces are the performance surfaces (BER surface) of wireless communications systems. 3.3.2 Procedure for Generating a Response Surface Model 3.3.2.1 Selection of an Appropriate Surface Model In response surface modeling, the first and most important step is to choose the model which will be used to model the response surface. Usually, two kinds of models can be chosen, specific empirical models and general models (polynomial models). Empirical models are always used when the shape of the response surface is similar to the shape of some empirical model. Empirical models are usually simpler than general models, because fewer parameters are needed in the empirical models. But in the case of BER surface modeling, it's difficult to find empirical models to fit B E R surface. So, general models (polynomial models) are used. Polynomial models can be further divided into quadratic and higher-order polynomial models. In quadratic models, the response surface is models as a second-order polynomial: 25 y = bo+ bixi + b2X2 + bjx^ + b4x2 + b&22 (3.1) The quadratic model is a very simple model. Six parameters are enough to present a quadratic model. But, in most cases, the surface is not so simple that can be modeled as a quadratic model. In such cases, a higher-order model can be used instead.. For example, the fifth-order polynomial model is shown in equation 3-2. 2 2 2 2 3 3 3 3 y=bo+b]Xi+b2X2+b3XiX2+b4Xi +b$X2 +bsXiX2 +67*/ X2+bsXi +bgX2 +bioXiX2 +buxi X2+bn Xi2X22+bi3Xi4+bi4X24+bj5XjX24+bj6Xi4X2+bi7Xi2X23+bi8Xj3X22+bi9X]5+b20X25 (3.2) For higher-order polynomial models, with the increasing of the polynomial order, the accuracy of the model is also increased. But at the same time, the model becomes more complicated. More parameters are needed to model the response surface for high order polynomial. For a third-order polynomial, 10 parameters are needed; for fourth-order polynomial, 15 parameters are needed; and for a fifth-order polynomial, 21 parameters are needed. Some transformations of outputs are always used (e.g., log, exp, square, square root, inverse) to aid model fitting and optimization. In the case of performance surface modeling of wireless communications systems, the log of the BER surface is taken in order to ease the modeling task. 3.3.2.2 Tools Used to Generate Response Surface Models The next step in generating a response surface model is to estimate the values of the parameters in the polynomial. For the case of modeling the B E R surfaces of wireless communications systems, the quadratic model is not adequate. So, higher-order polynomial models will be used. Higher-order polynomial models are more difficult to fit, requiring 26 iterative methods that start w i t h an in i t ia l guess o f the u n k n o w n parameters. E a c h iteration alters the current guess unt i l the a lgor i thm converges. In the Statistics toolbox o f M a t l a b , there are two functions that use the nonlinear least squares technique to fit a mode l that has a k n o w n parametric fo rm but u n k n o w n parameter values. The function nlinfit is designed for f inding parameters i n nonl inear mode l ing . Nlinfit returns the least squares parameter estimates. That is, it finds the parameters that m i n i m i z e the sum o f the squared differences between the observed response and their fitted values. It uses the Gauss -Newton a lgor i thm w i t h Levenberg-Marquard t modif ica t ions for g loba l convergence. The function nlintool is used to fit an equation to data and d isplay an interactive graph. The result o f nlintool is a predic t ion plot that provides a curve fit to the or ig ina l data set. It can plot an arbitrary g loba l confidence interval for predictions as two red curves. The default value for the confidence interval is 9 5 % . It 's very easy to estimate the values o f the parameters o f the response surface models and observe the fitting results by these functions. 3.3.3 Verification of the Accuracy of Response Surface Models 3.3.3.1 Verification Method F i n a l l y , the error o f the fitted mode l has to be analyzed to make sure the model is acceptable. U s u a l l y , lower-order po lynomia l s w i l l be used to fit the observed data set first. I f the errors produced b y the lower-order po lynomia l s are not acceptable, higher-order po lynomia ls should be used to improve the results. N o r m a l l y , the root mean square value is calculated for the residue (difference between estimated value f rom generated mode l and the observed data sets). The calcula t ion o f root mean square error is shown i n equation 3-3. 27 i x ; _ , g , - * , ) 2 rase=1,— — ( 3 3 ) In equation 3-3, xl is the estimated value from generated model, and x,is the observed value obtained from simulation or emulation, and n is the number of total data points. In the case of performance surface modeling of wireless communications systems, the log of the BER is used. For high BER (e.g., 10"1), a small error (e.g., 0.0001) is acceptable. But for low BER (e.g., 10"4), the same small error (0.0001) is a big problem. So the root mean square error which is calculated by absolute errors between estimated values and observed data sets is obviously not sufficient to evaluate the accuracy of the models. In this chapter, besides the absolute error, the error is also calculated as relative error compared with original data points. The calculation is shown in equation 3-4. , . error between estimated value and observed value relative error = (3.4) observed value Then the root mean square value is calculated for the relative errors. 3.3.3.2 First Example of Response Surface Modeling Two examples of response surface modeling are presented here. In the first example, a BER surface is generated corresponding to two channel parameters, RMS delay/symbol period and Maximum Doppler Frequency. The simulation model used to generate BER surface in Simulink is the same as the model shown in Figure 3-5. In this example, the wireless communication system is composed of a Random Binary Source, DBPSK modulation and demodulation blocks and channel. The channel includes Rician fading, Raleigh fading and A W G N . The symbol rate of the Binary Source is chosen as 2400 Hz. For the Rayleigh fading, the number of paths is five, and the Gain vector of the five paths is fixed as [0 -3 -6 -9 -12] dB. 28 The Rician K factor for the Rician fading is chosen as 30 dB, and the Es/No for the A W G N is chosen as 100 dB. These two values are chosen as high values to ignore the effect of these factors on the system performance. The results are shown in Table 3-1, 3-6 and Figure 3-7. Simulation Result 0.5 RMSdelay/T 0.5 RMSdelay/T RSM Result Polynomial order:2 300 0 0 (a) 200 100 Max Doppler Freq(Hz) RSM Result Polynomial order:3 0.5 RMSdelay/T 300 200 100 0 0 Max Doppler Freq (Hz) (b) RSM Result Polynomial order:4 0 0 (c) 200 100 Max Doppler Freq (Hz) 0.5 RMSdelay/T 0 0 300 200 100 Max Doppler Freq (Hz) (d) Fig 3-6 Comparison of BER surfaces (BER vs. RMS delay/T vs. Doppler spread for DBPSK modulation scheme) generated by simulation and response surface models, (a) Simulation result (b) Second-order polynomial model (c) Third-order polynomial model (d) Fourth-order polynomial model The BER surface obtained from Simulink simulation is shown in Fig 3-6 (a). The BER surface generated by quadratic polynomial is shown in Fig 3-6 (b). The BER surface generated 29 by third-order and fourth-order nonlinear polynomial are shown in Fig 3-6 (c) and (d) accordingly. It's obvious that the BER surface generated by quadratic model is not acceptable, and the order of the polynomial needs to be increased. When the order is increased to four, the surface looks quite similar with the result of simulation. The comparison of the shape of different surfaces is not adequate for assess the accuracy of the models. The RMSEs of absolute and relative errors are listed in Table 3-1 to compare the accuracy of various polynomial models. Table 3-1 Comparison of RMSE obtained from different response surface models (the polynomial models are obtained by fitting the surface of BER vs. RMS delay /T vs. Doppler spread with DBPSK modulation) Order of polynomial model Number of model parameters RMSE of absolute error RMSE of relative error (percentage) 2 6 0.148 8.5% 3 10 0.083 4.5% 4 15 0.052 2.8% Form Table 3-1, we can see that with the increase of polynomial order, the root mean square errors produced by the polynomial models decreases significantly. It is more clear to present the distributions of relative errors in a graph (Figure 3-7). It's easy to see from Figure 3-7 that with the increase of polynomial order, the range of errors decreases. The performance of second-order polynomial model is much worse than the other two models, and the fourth-order polynomial model is a littler better than the third-order polynomial model. Because the performance of fourth-order polynomial model is similar as the third-order polynomial model, there is no need to increase the polynomial order further. 30 0.3 0.2 0.1 -0.3 •0.4 second order third order fourth order 0 20 40 60 80 100 120 Fig 3-7 Distribution of relative errors obtained from different polynomial models (the polynomial models are obtained by fitting the surface of BER vs. RMS delay /T vs. Doppler spread with DBPSK modulation) 3.3.3.3 A Second Example of Response Surface Models Another example is presented here to show that response surface models can be used to fit any BER surfaces of various wireless communication systems. In this example, the BER surface is generated corresponding RMS delay/symbol period and Es/No. The wireless communication system is composed of a Random Binary Source, M S K modulation and demodulation blocks and channel. The channel also includes Rician fading, Rayleigh fading and A W G N . The parameter setting of Binary Source, Rayleigh fading is the same as last example. The Rician K factor for the Rician fading is chosen as 20 dB, and Maximum Doppler frequency of both Rayleigh and Rician fading is chosen as 50 Hz. Second, third, fourth and fifth polynomial models are used to fit the surface generated by simulation subsequently, and the results are shown in Figure 3-8, Figure 3-9 and Table 3-2. 31 Simulation Result 0.5 RMSdelay/T Es/No(dB) (a) 0.5 RMSdelay/T RSM Result Polynomial order:2 RSM Result Polynomial order:3 Es/No(dB) 0.5 RMSdelay/T RSM Result Polynomial order:4 RSM Result Polynomial order:5 Fig 3-8 Comparison of BER surfaces (BER vs. RMS delay/T vs. Es/No for M S K modulation scheme) generated by simulation and response surface models (a) Simulation result (b) Second-order polynomial model (c) Third-order polynomial model (d) Fourth-order polynomial model (e) Fifth order polynomial model The BER surface obtained from the Simulink simulation is shown in Fig 3-8 (a). The BER surface generated using a quadratic polynomial is shown in Fig 3-8 (b). The BER surfaces generated by third-order, fourth-order, and fifth order polynomial models are shown in Fig 3-8 (c), (d) and (e) accordingly. With the increase of the order of polynomial models, the surfaces generated by polynomial models better fit the data obtained from simulation. The RMSEs of absolute and relative errors are listed in Table 3-2, and the distributions of relative errors are shown in Figure 3-9 in order to compare the accuracy of various polynomial models. Table 3-2 Comparison of RMSE obtained from different response surface models (the polynomial models are obtained by fitting the surface of BER vs. RMS delay IT vs. Es/No with M S K modulation) Order of polynomial model Number of model parameters RMSE of absolute error RMSE of relative error (percentage) 2 6 0.087 7.8% 3 10 0.039 3.6% 4 15 0.023 1.9% 5 21 0.015 1.3% From both Table 3-2 and Figure 3-9, we can see that with the increase of polynomial order, the accuracy of the polynomial models also increase significantly. The performance of second-order and third-order polynomial models are much worse than the other two models, and the fifth-order polynomial model is a littler better than the fourth-order polynomial model. Because the performance of fourth-order polynomial model is quite similar as the fifth-order polynomial model, the order five is enough for the response surface model. 33 0.5 0.4 second order third order fourth order or > o 0.3 0.1 -0.2 0.1 0 0.2 0 50 100 150 200 250 Fig 3-9 Distribution of relative errors obtained from different polynomial models (the polynomial models are obtained by fitting the surface of BER vs. RMS delay /T vs. Es/No with M S K modulation) From the results of the two examples, it is apparent that the response surface models can be used to represent the BER surfaces of various wireless communication systems correctly and compactly. In the above examples, the BER surfaces are made up of 231 data points obtained from simulation, and at most fifth order polynomials are enough to correctly represent the BER surfaces. That means at most 21 parameters can present 231 data points correctly. 3.4 Implementation of Response Surface Models in Different Applications The response surface model is a very compact representation of the performance of wireless communication system. But at the same time, this representation also provides 34 complete information of the performance of wireless communication system. This efficient representation is very useful, and can be implemented in various applications, such as visualization of the performance of wireless communication system, network simulation, equipment testing, and in comparison of different equipment. 3.4.1 Application of Response Surface Models in Visualization of the Performance of Wireless Communication System As mentioned before, there is a useful function, nlintool, in the Statistics toolbox of Matlab. Nlintool is not only a tool for fitting and prediction for nonlinear models. This function also provides a graphic user interface (GUI), so the exploration of the graph of a multidimensional nonlinear function is easy to be done. Depending on the number of independent variable, the GUI provides a set of plots. Each plot shows the fitted relationship of the response surface to one independent variable at fixed values of other independent variables. The fixed value of each independent variable is in an editable text box below each axis. The fixed value of any independent variable can be changed easily. When the value of an independent variable is changed, all the plots update to show the current condition. In this chapter, only two channel parameters (independent variables) are used to generate performance surfaces of wireless communications systems. The fifth-order polynomial model generated from the second example in previous section is shown in Figure 3-10 to show the GUI provided by nlintool. In this example, the confidence intervals in chosen as 99%. The confidence interval for predictions is shown as two red curves in the plots. The predicted BER (10" 1 1 5 5 7 for this example) is shown at the left side of the graph. The pop-up menu labeled as Export can be used to move specified variables to the workspace of Matlab. Although in this example, only three parameters are shown in plots, nlintool can accommodate an arbitrary number of channel parameters. With the help of Matlab function 35 nlintool, the response surface model can be easily used to visualize the effect of any channel parameter on the performance of wireless communication systems. From the plots generated by nlintool, the BER value corresponding to any combination of channel parameters can be estimated according to the polynomial model. Fig 3-10 Performance of wireless communication system (BER vs. RMS delay/T vs. Es/No for M S K modulation scheme) show in nlintool GUI 3.4.2 Application of Response Surface Models in Network Simulation It has been mentioned in the introduction that current methods used to assess the performance of wireless communications systems suffer from significant limitations when applied to network level simulations. There are two main challenges: (1) predicting the link-36 level performance of a wireless communication system is computationally intensive and (2) the results usually involve a great deal of data. Response surface models can be used to overcome these limitations. First, the computationally intensive generation of the performance can be done by physical layer simulation. Then the results obtained from physical layer simulations can be further compressed as a function of various propagation channel parameters. This compact representation, which can be regarded as an equipment performance model, is easily implemented in network simulation as suggested by Fig. 3-11. Network Layout Usage Equipment List V i Propagation j Models i •'• Equipment | Performance j ! Usage Models 1 ; Models Wireless Performance Metrics Fig 3-11 Wireless performance prediction using equipment performance models (response surface model) 3.4.3 Application of Response Surface Models in Evaluation of the Performance of Equipment Response surface models can also be applied to experimental methods for assessing the performance of wireless communications systems based upon a test controller, RF test set and RF channel emulator. 37 In order to determine how propagation impairments affect the performance of an actual system, experiments should be performed using the real RF channel. Considering the cost, repeatability, accuracy, and completing testing over a full range of channel conditions, the Channel Emulator is a good choice to replace the real RF channel. A test-bed setup for conducting performance testing on real wireless devices and obtain system performance is suggested in Figure 3-12. BERT 4 RF TAS Channel RF Xmtr Emulator Rcvr under test under test Test Controller Fig 3-12 A test bed for conducting RF performance testing In the test bed, the TAS 4500 Channel Emulator transmits signals in the forward direction only. It is remotely controlled by the test controller through the GPIB-ENET 100 controller. The BERT (Bit Error Rate Test Set) is used to generate traffic and measure the performance of the devices under test. 38 During the test, the traffic generated from BERT is sent to the transmitter. The output is sent to the TAS4500 Channel Emulator. When the traffic passes through the TAS4500, some propagation impairments generated by the Channel Emulator distort the original data. Then the distorted data is received by the receiver. Finally, the BERT compares the transmitted and received traffic to calculate the system performance. Through GPIB-ENET controller, the value of the channel parameters can be changed dynamically, and the channel parameter sets and the corresponding performance can be recorded automatically. Finally, the BER surfaces can be plotted corresponding to the channel parameters. Then, the response surface models can be used to simplify the BER surfaces and give a compact representation. 3.4.4 Application of Response Surface Models in Comparing the Performance of Different Equipment If the values of the propagation channel parameters are specified, it's simple to estimate the BER performance according to the response surface models. It's also easy to predict the range of BER performance when the range of channel parameters is given. If a complete first-order statistical description of the propagation channel parameters applicable to a particular environment is available, the response model can be used to predict the probability of channel outage that would be encountered in that specified environment. Usually, performance comparison of different equipment under some particular environment is not adequate. The response surface models can provide the comparison of performance range and comparison of system outage probability for any environment. When the propagation conditions are changed, it's easy to use the same surface response model to predict the system performance. 39 3.5 Conclusion W e have shown that p o l y n o m i a l response surface models are a par t icular ly efficient way to complete ly describe the bit error rate ( B E R ) performance o f wireless communicat ions systems as a funct ion o f standard channel parameters. A fifth-order p o l y n o m i a l specified by twenty-one parameters can mode l a B E R surface based upon two channel parameters w i t h a root mean square error ( R M S E ) o f on ly a few percent. The M A T L A B function nlinfit easi ly generates such response surface models from either measured or s imulated data. Response surface models can be used i n many applicat ions. The M A T L A B function nlintool a l lows one to easi ly v i sua l ize response surface models that reveal the effect o f any propagation impairment on system performance. Because p o l y n o m i a l response surface models are compact, they can be convenient ly used as equipment performance models i n higher level system simulat ions and thereby make such simulat ions m u c h more real is t ic . The results presented here suggest that it is possible to develop a measurement-based mode l that comple te ly captures the l i nk - l eve l performance o f a wire less communica t ions system over a broad range o f channel condit ions based upon on ly a few tens o f measurements us ing a test setup that incorporates an R F channel emulator. W e recommend that the feasibi l i ty o f this approach be demonstrated at the earliest opportunity. The v i sua l i za t ion schemes presented in Sect ion 3.2 represent on ly the first step toward us ing response surface m o d e l l i n g for: (1) the v i sua l iza t ion o f the manner i n w h i c h channel parameters j o i n t l y affect equipment performance and (2) ident i f icat ion o f c o m m o n faults or defects i n the equipment based upon characteristic trends observed i n the response surface model . W e recommend that the u t i l i ty and feasibi l i ty o f these applicat ions be demonstrated at the earliest opportunity. 40 The utility of a response surface model increases as the representation becomes more compact. Reducing the size of the model becomes simpler i f the accuracy requirements for such models are not made more stringent than necessary. Accordingly, we recommend that steps be taken to determine reasonable accuracy requirements for such models in practical applications. The required sampling interval is determined by the complexity of the surface. If the response is sampled over a fixed grid, more points may be sampled than required in regions where the response surface is comparatively flat. This may lead to excessive simulation or measurement times. In order to reduce the effort required to generate a surface response model, we propose to adaptively sample the response instead. That is the subject of the next chapter. 41 C h a p t e r 4 U s e o f A d a p t i v e S a m p l i n g t o S i m p l i f y D e t e r m i n a t i o n o f t h e P e r f o r m a n c e o f W i r e l e s s C o m m u n i c a t i o n s S y s t e m s i n t h e P r e s e n c e o f P r o p a g a t i o n I m p a i r m e n t s 4.1 Introduction In Chapter Three, we introduced the use of polynomial response surface models to provide complete yet compact representations of the BER performance of wireless communications systems in the presence of propagation impairments. Such models are useful for the purposes of network-level simulation (using packages such as OPNET or ns2) and for comparing the performance of alternative wireless communications systems over a broad range of channel conditions. A response surface model is created by sampling the BER performance of a wireless communications system as a function of one or more propagation impairments over a broad range of channel conditions then fitting a polynomial surface in such a way that the root mean square error is minimized. The result is a complete yet compact representation of the performance of the system. In order to obtain accurate response surfaces, we must sample the response at a sufficient number of points. If the response is sampled over a fixed grid, more points may be sampled than required in regions where the response surface is comparatively flat. This may lead to excessive simulation or measurement times. In order to reduce the effort required to generate a surface response model, we propose to adaptively sample the response instead, as suggested by Fig. 4-1. 42 Fig 4-1 Use of adaptive methods to concentrate samples in regions with complicated profiles. Adaptive sampling was first introduced in 1962 by Dorf et al. [18] who used the constant integral-difference (ID) criterion to set the sampling rate. Similar adaptive sampling algorithm was proposed by Mitchell and McDaniel in 1969 [19]. The general approach has since been widely adopted with many kinds of adaptive sampling algorithms being used in variety of environments [20-25]. For the case of response surface modeling, we limit ourselves to the three-dimensional case in which two channel parameters are sampled and system performance (BER) is calculated. Our problem is to determine how to sample the channel parameter space less often while maintaining the correct shape of the response surface. A similar problem is encountered in image processing and many sophisticated adaptive sampling algorithms developed to solve this problem [26, 27, 28, 29]. For image processing, the aim of the adaptive sampling is to achieve the best quality of reconstructed image with fixed number of sample points. In 1991, Demetri Terzopoulos and M . Vasilesu proposed adaptive meshes to sample and reconstruct image [26]. In their method, the number of the sample points is fixed, and the position of the sample point is recalculated in each iteration to concentrate near 43 rapid shape variations. For response surface modeling, the simulation or experimental time should also be considered. Recalculating the position of the sample point requires that one double the simulation or experimental time. As a result, this kind of adaptive sampling is not appropriate here. To balance simulation or measurement time vs. response accuracy, the adaptive sampling algorithm must trade off complexity vs. efficiency. Although there are many kinds of adaptive sampling algorithms, none seemed appropriate for our case. To meet our requirements, we introduce the simple and efficient adaptive sampling algorithm described here. The remainder of this chapter is organized as follows: In Section 4.2, we introduce the concepts that form the basis for our adaptive sampling algorithm for response surface modeling. In Section 4.3, we describe how we implemented our adaptive sampling algorithm. In section 4.4, we discuss the performance of our adaptive sampling algorithm. In Section 4.5, we draw conclusions and offer recommendations for further work. 4.2 Concept Our adaptive sampling algorithm reduces the time required to characterize the response surface by sampling finely when response surface changes greatly, and sampling coarsely when the response surface changes slowly. It is a two-phase algorithm that is both simple and easy to implement. This two-phase sampling approach is motivated by Thompson's proposal for a two-phase adaptive design or adaptive cluster sampling (ACS) [21]. The idea of ACS is to take an initial sample by some ordinary sampling procedure, and then increase the sample size by adding samples in vicinity of the sampled units that satisfy a previously specified condition. In the first phase, the surface is coarsely sampled. The aim is to map the general shape of the surface rather than account for fine details. Because we have constrained ourselves to dealing with three-dimensional surfaces, we can connect each sample point to its natural 44 neighbours in order to create a Delaunay triangulation. When the first phase is complete, we have resolved the surface into a continuous set of triangular plates. In the second phase, each triangular plate is subdivided into smaller triangular plates. If angular difference between the normal vectors to the original plate and each of the sub-plates are sufficiently large, we break a plate into smaller plates that follow the surface more closely. In this phase, the maximum number of divisions is set at the beginning in order to avoid infinite division when response surface changes suddenly and dramatically. This process of subdivision is done repeatedly until all the sub-plates either satisfy the criterion or the maximum number of divisions is reached. The preceding explanation of phase two is only a brief overview. The details are as follows. The criterion used to decide if a division is needed for one plate is the angles between the original triangular plate and the two sub-triangular plates which are obtained by dividing the original triangular plate. One example is shown in Fig 4-2. P, P 2 P 3 Fig 4-2 The criterion to decide another division As shown in Fig 4-2, one triangle (P1P2P3) is divided into two sub-triangles (P1P2P4 and P3P2P4). Then A g l , the angle between the triangles P1P2P3 and P1P2P4, is calculated and 45 compared with the threshold 1 (Ag). The same is done to Ag2, the angle between the triangles PiP2P3and P3P2P4- If any of the two angles (Agl and Ag2) is bigger than the threshold 1 (Ag), the original triangle P1P2P3 is not small enough and needs further division, then the two sub-triangles P3P2P4and P1P2P4 will be put into the pool and wait for next division. It can be seen that either of the two angles which is greater than the threshold 1 (Ag) will change the response surface greatly and the original triangle must be subdivided. The angle between two triangles can be obtained by calculating the angle between the normal vectors of the two triangles. As our last example, the angle A g l can be obtained by the following steps: Step 1: Calculate the normal vector (NormO) of the first triangle P 1 P 2 P 3 P2P1 = [iM*)-P,(x), P2(y)-Pi(y), P2(z)-P,(z)] (4.1) P2P3 = [P2(x)-P3(x), P2(y)-P3(y), P2(z)-P3(z)] (4.2) NormO = P2Pi x P2P3 (4.3) Step 2: Calculate the normal vector (Norml) of the second triangle P3P2P4 according to equation 4.1, 4.2 and 4.3 Step 3: Calculate the angle between the two Normal Vectors (Agl) NormO • Norml A g l = arccos (4.4) I NormO || Norml \ The adaptive sampling portion is based on the basic units, triangles generated from coarse fixed rate sampling. For each unit, the decision is made for all triangles including sub-triangles whether another division is needed. There is another threshold, maximum number of divisions (NdO), which is also used when deciding when a particular unit should be further divided. When 46 there is some sudden or dramatic change in the response surface, division will have to be done many times to smooth the sudden change. To avoid unnecessary division, a maximum number of divisions (NdO) is specified. When all the triangles meet the requirement, or the maximum number of divisions has occurred, no further division is allowed for this unit. The same process is repeated for all of the units. The entire process of the adaptive sampling algorithm is summarized in Fig 4-3. How to choose the two thresholds is very important in order to make the adaptive sampling efficient. The first threshold, the angle between two triangles (Ag), is usually chosen between 0.5 and 1 degree. As the parameter Ag becomes larger, the accuracy of the response surface decreases but so does the simulation time. The second threshold is maximum number of divisions (NdO). Here, it is usually chosen as 2n-l. As the parameter NdO increases, the accuracy of the response surface increases as does the simulation time. The simulation time and accuracy has to be traded off when choosing the two thresholds. 4.3 Implementation Before we implemented the adaptive sampling algorithm for use in response surface modeling, we tested the algorithm using a simple function. In the test, two parameters x and y and their response z are considered. Here, z is obtained by equation 4-5. z = e - x * e°- 4* y (4.5) At first, fixed rate sampling is done to form the coarse response surface. The sampling numbers for parameters x andy are 11 and 16. x is chosen from 0 to 10 andjy is chosen from 0 to 15. After coarse sampling, the number of total data points is 176. Then adaptive sampling is performed. The angle threshold Ag is chosen as 1 degree and the maximum division time NdO is chosen as 7. The total data points after adaptive sampling is 396. 47 start J^) Coarsely sampling Ns: total point after cosa Nt: total triangles after cosa i K ^ ^ F o r i=1:1:Nt: Nd:division time=0 Put the triangle i into pool, Ntp: the number of triangles in the pool : Select one triangle from the pool id: the triangle be selected = 1 Divide triangle id into two sub-triangles and record the coordinates of the dividing point; Nd=Nd+1; Ns=Ns+1 Get the BER by simulation Get the coordinates of this point; Ns = Ns -1 Calculate the angles between triangle id and two sub-triangles: Ag1, Ag2 Put the two sub-triangles into the pool waiting for next division; Ntp = Ntp+2 Fig 4-3 Flow chart of Adaptive Sampling To validate the algorithm, three fixed rate sampling tests were also done to compare with the adaptive sampling test. The number of samples of parameters x and y for the three tests are: 16 and 25, 26 and 39, and 41 and 61. The three fixed rate sampling tests were designed so that one of them used the same total number of points as the adaptive sampling test; the other one achieves the same R M S E as the adaptive sampling test. The R M S E is obtained by comparing the adaptive sampling test or fixed rate sampling tests with the third fixed rate (a high-rate) sampling test. The R M S E can be estimated by 3 steps. Step 1: Get each sampling point in the high-rate fixed rate sampling test. Step 2: Search the point in the test compared with the high-rate fixed rate sampling test. (It can be either an adaptive sampling test or fixed rate sampling tests. For convenience, this test is described as testO and the high-rate fixed rate sampling test is described as testl later.) Step 3: If the point is one of the points of testO, then the error is 0. Else, assume this point is in one of the triangles of testO, and then use the x, y coordinates of this point and the coordinates of the triangle, the z coordinate of the point can be calculated. Then the error can be obtained by calculating the difference between the calculated and original z coordinates. In step three, i f the point is not one of the points of testO, the triangle containing this point has to be chosen first. To find the triangle that contains the point, all the triangles must be tested. The criterion used here is that i f one point is in one triangle, then the sum of the angles between this point and three vertices of the triangle should be 180 degree. After the right triangle is found, the z coordinate can be calculated. We assume the vertices of the triangle are A, B and C, and the point is D. Then the z coordinate of point D can be calculated as equations (4.6) and (4.7). 49 Nvector = AB * BC (4.6) D(z) = A(z) - N v e c t o r W x ( ^ ) ~ A(*)) + Nvector(y) x (D(y) - A(y)) Nvector(z) Similar steps are performed when adaptive sampling is used to generate performance surfaces for wireless communication systems. Two sets of experiments were conducted to verify the performance of adaptive sampling algorithm when it is used in response surface modeling. First, the effect of Rician K-factor and RMS delay/T (T is symbol period of the transmitted signal) on the performance of a wireless communication system based upon a DBPSK modulation scheme was considered. In the second set of experiments, the effect of Maximum Doppler Frequency and RMS delay/T on the performance of a wireless communication system with DQPSK modulation scheme was considered. In each set of experiments, five kinds of sampling methods were used. First, the coarse sampling was done. Then adaptive sampling was performed on the basis of the coarse sampling. Finally, three different fixed-rate sampling approaches were done to compare with the adaptive sampling. The three fixed rate sampling tests were also designed so that one of the fixed rate sampling tests used the similar total number of points as the adaptive sampling test, and the other test achieved the similar RMSE as the adaptive sampling test. The R M S E was obtained by comparing the adaptive sampling test or fixed rate sampling test with the third high-rate fixed rate sampling test. 4.4 Performance The performance of the adaptive sampling algorithm when applied to a simple function is presented using three approaches. First, the response surfaces of the three kinds of sampling 50 approaches are put together to compare their performance. Next the total number of generated points and the RMS error for the three kinds of sampling approaches are listed in tables. Finally, the column charts allow us to compare the adaptive sampling and fixed rate sampling approaches. The function to be sampled has to satisfy the requirement that part of the response surface changes quickly and part of the response surface changes slowly. Equation (4.4) is chosen because it meets this requirement but is also simple. The comparison of response surfaces generated by different sampling methods is shown in Fig 4-4. In Fig 4-4, it can be seen that fixed rate sampling uses too many points on the parts where the surface changes little. Comparing Fig 4-4 (b) with Fig 4-4 (c), we know when the total sampling points are the same for both sampling schemes, the fixed rate sampling approach samples the surface evenly, and adaptive sampling algorithm concentrate its sampling on the part where the surface changes quickly. When comparing Fig 4-4 (b) and Fig 4-4 (d), we see that when the adaptive sampling algorithm has the same sampling accuracy (RMSE) as fixed rate sampling approach, the adaptive sampling algorithm uses 25% less sampling points. N 0.54 0. 0 0 x 10 15 y X 10 15 y (a) (b) 51 Fig 4-4 Comparison of adaptive and Fixed rate sampling when they are tested by function (a) Coarsely sampled (b) Adaptive sampled (c) Fixed-rate sampled 1 (d) Fixed-rate sampled 2 Two sets of experiments were conducted in order to compare the performance of adaptive sampling and fixed rate sampling. The results are shown in Table 4-1,2 and Figs. 4-5 and 4-6. The first sets of experiments are designed to compare the R M S E of the two sampling methods when the total number of sampling points are the same. The results are shown in Table 4-1 and Fig 4-5. Table 4-1 RMSE comparison of adaptive and fixed rate sampling when they are tested by a simple function Sampling method Total generated points RMS Error compared with Fixed rate sampling (Sampling rate: 41*61 ) Coarsely Sampled 176 0.008 Adaptive Sampling Threshold: 1 degree 396 0.0012 Threshold: 0.03 degree 655 0.0011 Fixed rate Sampling Sampling rate: 16*25 400 0.0031 Sampling rate: 21*31 651 0.0021 Sampling rate: 41*61 2501 0 * For the Adaptive Sampling, the maximum number divisions is 7. 52 • Adaptive Samplig H Fixed rate Sampling 0.0035 The number of totoal points: The number of totoal points: about 400 about 650 Fig 4-5 RMSE comparison of adaptive and fixed rate sampling when they are tested by a function This set of experiments includes two adaptive sampling tests and three fixed rate sampling tests. The maximum number of divisions (NdO) for both the adaptive sampling tests is 7, and the angle thresholds (Ag) are 1 and 0.03 degree. The sampling rate for the high rate fixed rate sampling is: x = 41, y = 61; the sampling rates for the other two fixed rate sampling are: x = 16, y = 25; x = 21, y = 31. From the column chart (Fig 4-5), one can see that when the total data points are same for both of the sampling methods, the RMSE of adaptive sampling algorithm is much better than the fixed rate sampling approach. When the number of total data points is 400, the RMSE of adaptive sampling is only 38.7% of that of fixed rate sampling. When the number of total data points is 650, the RMSE of adaptive sampling is approximate half of the RMSE of fixed rate sampling. 53 Another set of experiments were done to compare the total number of data points when the RMSE are same for both of the sampling approaches. Three adaptive sampling and three fixed rate sampling tests were conducted. The results are shown in Table 4-2 and Fig 4-6. The maximum number of divisions (NdO) for all the adaptive sampling tests is still 7. The angle thresholds (Ag) are: 6, 2, and 1 degree. The sampling rates for the three fixed rate sampling schemes include: x = 18, y = 27; x = 22,y = 33; and x - 26, y = 39. From Table 4-2 and Fig 4-6, we can see that when the two sampling methods achieve the same RMSE, the adaptive sampling algorithm can save a lot of sampling points compared to the fixed rate sampling approach. When the RMSE is 0.0023, the adaptive sampling uses 30% less data points; when the R M S E is 0.0016, the adaptive sampling uses 50% less data points and when the R M S E is 0.0012, the adaptive sampling uses 60% less data points. The more demanding the requirement for the RMSE, the greater is the advantage of the adaptive sampling algorithm over the fixed rate sampling approach. Table 4-2 Total data points' comparison of adaptive and fixed rate sampling when tested by function Sampling method threshold (degree) for adaptive sampling or sampling rate for Fixed rate sampling Total generated points RMS Error compared with Fixed rate sampling (Rate 3) Adaptive Sampling 1 6 340 0.0023 Fixed rate Sampling 1 18*27 486 0.0023 Adaptive Sampling 2 2 368 0.0016 Fixed rate Sampling 2 22*33 726 0.0016 Adaptive Sampling 3 1 396 0.0012 Fixed rate Sampling 3 26*39 1014 0.0012 For the Adaptive Sampling, the maximum number of division is 7. 54 • Adaptive Samplig • Fixed rate Sampling 1200 1000 | 800 o CL CD co "co P 400 600 200 0 RMS Error = 0.023 RMS Error = 0.016 RMS Error = 0.012 Fig 4-6 Total data points comparison of adaptive and fixed rate sampling when tested by a simple function In this project, the adaptive sampling algorithm is designed to generate BER surfaces for response surface modeling. So, two sets of tests were conducted to validate the performance of the adaptive sampling algorithm when it is used to generate BER surfaces. At first, a wireless communication system with DBPSK modulation scheme is considered, and Rician K-factor and RMS delay/ symbol period (T) are chosen as channel parameters. The model used to simulate the wireless communication system is the same as the model shown in Fig 3-5. The parameter settings for the above model are: simulation time is 50s; the symbol rate of the transmitted signal is 2400 Hz; Maximum Doppler spread for the fading channel is 50 Hz; Es/No for the A W G N channel is 100 dB; The Rayleigh fading channel is composed of 5 paths. The gain for each path is fixed as: 0, -3, -6, -9, -12 dB. The performance comparison between 55 adaptive sampling and fixed rate sampling are still represented by three ways. First, the BER surfaces are generated by different sampling methods are all shown in Fig 4-7. RMSdelay/T Sampling rate: 6*6 Simulation time: 5.7min Threshold; 1 degree Max number of divisons: 9 Simulation time: 49 min RMSE: 0.0094 K-factor (dB) K-factor (dB) RMSdelay/T Sampling rate: 18*18 Simtilationtime: 49 min RMSE: 0.0121 0 0 K-factor (dB) (c) O r " ' " RMSdelay/T .Sampling, rate: 21*20 Simulation time: 64.9 min RMSE: 0.0093 K-factor (dB) (d) Fig 4-7 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate BER vs. K vs. RMS delay/T for DBPSK modulation (a) Coarsely sampled (b) Adaptive sampled (c) Fixed-rate sampled 1 (d) Fixed-rate sampled 2 In Fig 4-7, the coarse sampling rate is chosen as 6 samples for both K-factor and RMS delay/T. The maximum number of divisions and angle threshold for adaptive sampling are 56 chosen as 9 and 1 degree. A high rate fixed rate sampling (21 samples for both K-factor and RMS delay/T) was conducted as the basis to calculate the RMSE. Two other rate fixed rate sampling is done to satisfy that one achieves similar RMSE and one spends similar simulation time as adaptive sampling. When we compare Fig 4-7 (b) and Fig 4-7 (c), we see that when the two sampling methods spend similar simulation time, adaptive sampling yields more accurate results than fixed rate sampling. When the Fig 4-7 (b) is compared with Fig 4-7 (d), we see that adaptive sampling requires much less simulation time and obtains similar RMSE as fixed rate sampling. The result is also shown in Table 4-3. Table 4-3 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate BER vs. K vs. RMS delay/T for DBPSK modulation Sampling method Total generated points Total simulation time (minute) RMS Error compared with Fixed rate sampling (Rate 3) Coarsely Sampled 36 5.7 0.0581 Adaptive Sampling 324 49 0.0094 Fixed rate Sampling Rate 1 324 49 0.0121 Rate 2 420 64.9 0.0094 Rate 3 441 66.4 0 * For the Adaptive Sampling, the threshold is 1 degree and maximum number of divisions is 9. For the Fixed rate Sampling, the Rate 1 is 18*18 (K*RMS delay/T), Rate 2 is 21*20 and Rate 3 is 21*21. From Table 4-3, we see that when the total simulation time is 49 minutes, the RMSE for the adaptive sampling is 77.7% of that for fixed rate sampling. When the RMSE is 0.0094 for both of the sampling methods, the adaptive sampling saves 25% simulation time. Another set of similar experiments is also done to verify the generality of the adaptive sampling algorithm. Here, the model of the wireless communication system is similar as before, except that the DBPSK modulation and demodulation scheme is replaced by DQPSK 57 modulation and demodulation scheme; and RMS delay/T and Maximum Doppler Frequency are chosen as channel parameters. The parameter settings for the system model are: simulation time is 50s; the symbol rate of the transmitted signal is 2400 Hz; Rician K-factor is 30 dB; Es/No for the A W G N channel is 100 dB; the gains for each path of the Rayleigh fading channel are: 0, -3, -6, -9, -12 dB. The performance comparison is shown in Fig 4-8 and Table 4-4. RMSdelay /T Sampling rate: 6*6 Simulation time: 5.7 min Threshold: 0.8 degree Max number of divisions: 7 Simulation time: 38.6 min R M S E : 0.0125 \ 0 0 Max Doppler Freq (Hz (a) 0.5 R M S d e l a y / T 0 0 Max Doppler Freq (Hz) (b) 0.5 RMSdelay /T Sampling rate: 16*16 Simulation time: 39.2 min R M S E : 0.0137 a: co -2 o Sampling rate: 20*18 Simulation time: 56 min R M S E : 0.0124 150 100 0 5 50 0 0 Max Doppler Freq (Hz) RMSdelay/T 0 0 Max Doppler Freq (Hz) (c) (d) Fig 4-8 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate BER vs. Doppler spread vs. RMS delay/T for DQPSK modulation (a) Coarsely sampled (b) Adaptive sampled (c) Fixed-rate sampled 1 (d) Fixed-rate sampled 2 58 Fig 4-8 shows the comparison of BER surfaces generated by different sampling methods. The adaptive sampling is obviously better than fixed rate sampling both when they achieve same RMSE and when they spend same simulation time. Numerical comparison of these two sampling methods can be found in Table 4-4. Table 4-4 Comparison of adaptive and fixed rate sampling when they are tested in the simulations to generate BER vs. Doppler spread vs. RMS delay/T for DQPSK modulation Sampling method Total generated points Total simulation time(minute) RMS Error compared with Fixed rate Sampling (Rate 3) Coarsely Sampled 36 5.7 0.7657 Adaptive Sampling 238 38.6 0.0125 Fixed rate Sampling Rate 1 256 39.2 0.0137 Rate 2 360 56 0.0124 Rate 3 1681 269 0 * For the Adaptive Sampling, the threshold is 0.8 degree and maximum number of divisions is 7. For the Fixed rate Sampling, the Rate 1 is 18*18 (Maximum Doppler Frequency*RMS delay/T), Rate 2 is 21*20 and Rate 3 is 21*21. From Table 4-4, one can see that when both of the sampling methods spend about 39 minutes, the RMSE for adaptive sampling is 90% of that for fixed rate sampling. And when they obtain similar R M S E (0.0125), the adaptive sampling reduces simulation time by 31%. The comparisons are also represented by a column chart. Fig 4-9 shows the comparison of simulation times when both of the sampling methods have similar RMSE. Fig 4-10 shows RMSE comparison when both sampling methods spend similar simulation time. From Fig 4-9 and Fig 4-10, it is apparent that adaptive sampling algorithm is much better than fixed rate sampling for both of the wireless communication systems considered here. 59 • Adaptive Samplig • Fixed rate Sampling DBPSK: BER vs. Rician K-facotor vs. DQPSK: B E R vs. Doppler spread vs. R M S delay/T RMS Error is about RMS dealy/T R M S Error is about 0.094 0.0124 Fig 4-9 Simulation time comparison of adaptive and fixed rate sampling when the sampling algorithms are tested in simulation D B P S K : B E R vs. Rician K-facotr D Q P S K : B E R vs. Doppler spread vs. R M S delay/T Total simulation vs. R M S dealy/T Totoal simultion time is about 49 minutes time is about 39 minutes F ig 4-10 R M S E comparison of adaptive and fixed rate sampling when the sampling algorithms are tested in simulation 4.5 Conclusion In this chapter, we have proposed and evaluated an adaptive sampling scheme that can significantly reduce the time and effort required to generate a B E R surface response model from physical layer simulations. In two typical scenarios, the adaptive sampling method was shown to achieve the same R M S E with between one-quarter and one-third less simulation time compared to traditional schemes that sample over a regular grid. Alternatively, the adaptive sampling method was shown to reduce the RMSE by between one-tenth and one-quarter given the same simulation time. Although these results were generated by simulating two typical wireless communication system scenarios, the technique is also applicable to experimental methods that capture the link-level performance of a system using a test setup that incorporates an RF channel emulator. The adaptive sampling scheme presented here represents only the first step toward improving the efficiency of response surface generation. In the limit, one should be able to estimate the n parameters of the response surface model using n uncorrected measurements. We recommend that steps be taken to develop suitably clever sampling schemes that will allow us to more closely approach this limit. The adaptive sampling scheme presented here is only applicable to three-dimensional surfaces that represent the dependence of system performance on two channel parameters. We recommend that steps be taken to develop adaptive sampling schemes that are applicable to higher dimensional B E R surfaces that represent the dependence of system performance on three or more channel parameters. 61 C h a p t e r 5 C o n c l u s i o n s a n d R e c o m m e n d a t i o n s 5.1 Conclusions This thesis has been concerned wi th the development o f efficient strategies for representing and evaluat ing the effect o f propagation impairments on the performance o f wireless communica t ions systems. Current methods for assessing the effect o f propagation impairments on l ink - l eve l performance either y i e l d results that apply on ly to certain specif ic channel condit ions or only produce statistical summaries o f l i nk - l eve l performance over a part icular set o f channel condit ions. W h i l e adequate for some purposes, a more complete descr ipt ion is required for applications such as system-level s imulat ion or v i sua l iza t ion o f l i n k - l e v e l performance. W e have shown that p o l y n o m i a l response surface models are a par t icular ly efficient way to comple te ly describe the bit error rate ( B E R ) performance o f wireless communicat ions systems as a funct ion o f standard channel parameters. A fifth-order p o l y n o m i a l specified by twenty-one parameters can mode l a B E R surface based upon two channel parameters w i th a root mean square error ( R M S E ) o f on ly a few percent. The M A T L A B function nlinfit easily generates such response surface models f rom either measured o r s imula ted data. Response surface models can be used i n many applications. The M A T L A B funct ion nlintool a l lows one to easi ly v i sua l ize response surface models that reveal the effect o f any propagation impairment on system performance. Because p o l y n o m i a l response surface models are compact, they can be convenient ly used as equipment performance models i n higher leve l system simulat ions and thereby make such s imulat ions m u c h more realist ic. 62 W e have also shown that adaptive sampl ing can be used to s igni f icant ly reduce the t ime and effort required to generate a surface response mode l f rom phys ica l layer simulat ions. The technique can be used either to improve the accuracy o f the response surface or to s ignif icant ly reduce the t ime required to generate a response surface. C o m p a r e d to tradit ional schemes that sample over a regular g r id , the adaptive sampl ing method can achieve the same R M S E wi th between one-quarter and one-third less s imula t ion t ime. Al t e rna t ive ly , the adaptive sampl ing method can reduce the R M S E by between one-tenth and one-quarter g i v e n the same s imulat ion t ime. 5.2 Recommendations for Further Work W h i l e this study has demonstrated that it is both feasible and pract ical to use p o l y n o m i a l response surfaces to m o d e l the B E R performance o f wireless communica t ions systems, we recommend that several topics o f considerable pract ical interest be pursued in the very near future: • Demonstration of the experimental determination of a response surface model. The results presented here suggest that it is possible to develop a measurement-based mode l that comple te ly captures the l i nk - l eve l performance o f a wireless communicat ions system over a broad range o f channel condit ions based upon on ly a few tens o f measurements us ing a test setup that incorporates an R F channel emulator. W e recommend that the feasibi l i ty o f this approach be demonstrated at the earliest opportunity. • Assessment of the potential role of response surface models for visualization and diagnostic purposes. The v isua l iza t ion schemes presented i n Chapter 3 represent o n l y the first step toward us ing response surface m o d e l l i n g for: (1) the v i sua l iza t ion o f the 63 manner in which channel parameters jointly affect equipment performance and (2) identification of common faults or defects in the equipment based upon characteristic trends observed in the response surface model. We recommend that the utility and feasibility of these applications be demonstrated at the earliest opportunity. Determine the accuracy requirements for response surface models. The utility of a response surface model increases as the representation becomes more compact. Reducing the size of the model becomes simpler i f the accuracy requirements for such models are not made more stringent than necessary. Accordingly, we recommend that steps be taken to determine reasonable accuracy requirements for such models in practical applications. Improve the efficiency of the sampling scheme. The adaptive sampling scheme presented in Chapter 4 represents only the first step toward improving the efficiency of the response surface model generation. In principle, one should be able to estimate the n parameters of the response surface model using exactly n measurements. We recommend that steps be taken to develop suitably clever sampling schemes that will allow us to approach this limit. Extension of adaptive sampling to greater than three dimensions. The adaptive sampling scheme presented in Chapter 4 is only applicable to three-dimensional surfaces that represent the dependence of system performance on two channel parameters. We recommend that steps be taken to develop adaptive sampling schemes that are applicable to higher dimensional BER surfaces that represent the dependence of system performance on three or more channel parameters. 64 References [I] M . Emshwiller, "Characterization of the performance of PSK digital radio transmission in the presence of multipath fading," Conf. Record, ICC 78, Paper 47.3, June. 1978. [2] V . K . Prabhu and L . J. Greenstein, "Analysis of multipath outage with applications to 90Mb/s PSK systems at 6 and 11 GHz," Conf. Record ICC 78, Paper 47.2, June 1978. [3] W. C. Jakes, Jr., "An approximate method to estimate an upper bound on the effect of multipath delay distortion on digital transmission," Conf. Record ICC 78, Paper 47.1, June. 1978. [4] L . J. Greenstein and V . K . Prabhu, "Analysis of multipath outage with applications to 90Mb/s PSK systems at 6 and 11 GHz," IEEE Trans. Commun., vol. COM-27, no. 1, pp. 68-75, Jan. 1979. [5] W. C. Jakes, Jr., "An approximate method to estimate an upper bound on the effect of multipath delay distortion on digital transmission," IEEE Trans. Commun., vol. COM-27, no. 1, pp. 76-81, Jan. 1979. [6] W. D. Rummler, " A multipath channel model for line-of-sight digital radio systems," Conf. Record, ICC 78, Paper 47.5, June. 1978. [7] W. D. Rummler, " A new selective fading model: application to propagation data," Bell Syst. Tech. J., vol. 58, no. 5, May-June. 1979. [8] W. D. Rummler, " A simplified method for the laboratory determination of multipath outage of digital radios in the presence of thermal noise," IEEE Trans. Commun., vol. COM-30, no. 3, pp. 487-494, Jan. 1982. [9] C. W. Anderson, S . Barber, and R. Patel, "The effect of selective fading on digital radio," Conf. Record ICC, vol. 2, pp. 33.5.1-6, 1978. [10] L . J. Greenstein and B. A . Czekaj, " A polynomial model for multipath fading channel responses," Bell Syst. Tech. J., vol. 59, no. 7, September. 1980. [II] D. R. Smith and J. J. Cormack, "Measurements and characterization of a multipath fading channel for application to digital radio links," Conf. Record ICC '82, vol. 3, pp. 7B.4.1-6, 1982. [12] L. J. Greenstein and M . Shafi, "Outage calculation methods for microwave digital radio," IEEE Trans. Commun., vol. COM-25, no. 2, pp. 30-39, Feb. 1987. [13] C-L. Liu and K . Feher, "Performance of non-coherent K / 4 - Q P S K in a frequency-selective fast Rayleigh fading channel," Proc, ICC, vol. 2, pp. 335.7.1-5, Apr. 1990. 65 [14] C-L. Liu and K . Feher, "Bit error rate performance of 7t/4-QPSK in a frequency-selective fast Rayleigh fading channel," IEEE Trans. Vehic. Techol, vol. VT-40, no. 3, pp. 558-568, Aug. 1991. [15] V . Fung, " B E R simulation of 7t/4-DQPSK in flat and frequency-selective fading mobile radio channels with real time applications," Conf. Record, ICC '91, vol. 2, pp. 553-557, 1991. [16] V . Fung, T. S. Rappaport and B. Thoma, "Bit error simulation for 7i/4-DQPSK mobile radio communications using two-ray and measurement-base impulse response models," IEEE J. Select. Areas commun., vol. 11, pp. 393-405, Apr. 1993. [17] Khuri, A . I., and Cornell, J. A. , (1987), Response Surfaces, Marcel Dekker, New York, N Y . [17.1] Myers, R. H. , and Montgomery, D. C , (1995), Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New York, N Y . [18] R. C. Dorf, M . C. Farren, and C. A . Phillips, "Adaptive sampling frequency for sampled-data control systems," IRE Trans. Automat. Contr., Vol . AC-7, pp. 38-47, Jan. 1962. [19] J. R. Mitchell and W. L. McDaniel, "Adaptive sampling technique," IEEE Trans. Automat. Contr. (Corresp.), Vol . AC-14, pp. 200-201, Apr. 1969. [20] A . Mostafa, and M . El-Hagry, "Adaptive sampling of speech signals," IEEE Trans. Commun., vol. COM-22, no. 9, pp. 1189-1194, Sep. 1974. [21] S. K . Thompson, "Adaptive cluster sampling," J Am Stat Assoc., Vol . 85, pp. 1050-1059, 1990. [22] G. Lifiana, M . Fernando, and P. Gianfranco, " E C G Data Compression Through Adaptive Sampling And Arithmetic Coding," Proc. Ann Int Conf. IEEE. Eng. Med. Biol. Soc, vol. 12, pp. 451-452, Nov. 1990. [23] J. S. Stadler, and S. Roy, "Adaptive importance sampling [digital communication]," IEEE J. Select. Areas commun., vol. 11, no. 3, pp. 309-316, Apr. 1993. [24] W. W. Kuo, R. Akella, and D. Fletcher, "Adaptive sampling for effective multi-layer defect monitoring," IEEE Symp. on Semic. Manuf, vol. 6-8, pp. D1-D4, Oct. 1997. [25] G. Debunne, M . Desbrun, M . Cani, and A . H. Barr, "Dynamic real-time deformations using space and time adaptive sampling," proc, Comput. Graph., Aug. 2001. [26] D. Terzopoulos, and M . Vasilescu, "Sampling and reconstruction with adaptive meshes," proc. IEEE CVRP. Conf, pp. 70-75, June. 1991. [27] C. - H . Lee, "Image surface approximation with irregular samples," IEEE Trans. Pattern Aanl. Anal. Machine Intell, vol. 11, no. 2, pp. 206-212, 1989. [28] D. G. McCaughey, and H . C. Andrews, "Image approximation by variable knot bicubic splines," IEEE Trans. Pattern Anal. Machine Intell, vol. 3, no. 3, pp. 299-310, 1981. 66 [29] I. We i s s , "Shape reconstruction on a va ry ing mesh ," IEEE Trans. Pattern Aanl. Anal. Machine Intell, v o l . 12, no. 4, pp. 345-362, 1990. 67 

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