Identification of Nonlinear Interference Sources with the Use of the
Contents
1. 1997 pp 269 274 2 D D Weiner Nonlinear Interference Effects in EMC Supplement to the Proc of 10 Int Zurich Symp On EMC Zurich March 1993 pp 114 127 3 S L Loyka and V I Mordachev Mathematical models and algorithms of electromagnetic compatibility analysis and prediction software complex Technical report vol 1 Belorussian State University of Informatics and Radioelectronics Minsk Belarus 1997 4 V I Mordachev Express analysis of electromagnetic compatibility of radioelectronic equipment with use of discrete models of interference and fast Fourier transform Proc of IX Inter Wroclaw Symp on EMC Poland Wroclaw 1988 Part 2 pp 565 570 5 S L Loyka V I Mordachev Computer aided nonlinear simulation at the system level Proc of 5 Inter Confer On EMC EMI INCEMIC 97 Hyderabad India Dec 3 5 1997 pp 93 98 6 S W Chen W Panton and R Gilmore Effects of Nonlinear Distortion on CDMA Communication Systems IEEE Trans On MTT vol 44 No 12 Dec 1996 pp 2743 2750 7 B Gallagher Estimating and Measuring C I in a GSM Wireless Local Loop Receiver Microwave Journal vol 40 No 10 Oct 1997 pp 70 83 8 J Staudinger Applying the Quadrature Modeling Technique to Wireless Power Amplifiers Microwave Journal vol 40 No 11 Nov 1997 pp 66 86 9 J L Bogdanor R A Pearlman and M D Sieyel Intrasystem Electromagnetic Compatibility Analysis Program Volume I User s Manu2. signals M and J internal variables DICHOTOMOUS SEARCH ALGORITHM The dichotomous search algorithm based on the method given in the previous section are presented in Figure 4 Let s now consider the main steps of the algorithm After the start all signals are sorted according to the increase in power Then the user chooses an interference to identify and the current set of signals is equated to all the signals After that the procedure DIVIDE_TWO is called The main function of this procedure is to divide the current signal set CSS into two parts CSS and CSS in such a way that the first part contains smaller signals and the second one lager signals to turn off the first part to conduct the analysis and to check whether the interference disappeared If it did not disappear then the turned off part is excluded from the current signal set and the process of division is continued If interference disappeared then the second part CSS2 is turned off the first part remains to be turned on and the analysis is repeated If the interference did not disappear then the interference source is in the first part only and the process of division is continued for this part the second part is excluded from further consideration If interference disappeared then the second part contains interference sources In this case the procedure DIVIDE _TWO executes a series of internal settings and checks whether the number of interference signal
3. IEEE International Symposium on Electromagnetic Compatibility Denver Colorado Aug 24 28 1998 pp 882 887 Identification of Nonlinear Interference Sources with the Use of the Discrete Technique Sergey Loyka Belorussian State University of Informatics amp Radioelectronics Brovki Str 6 Minsk 220027 BELARUS Abstract This paper deals with a dichotomous method for the computer aided search of nonlinear interference sources in complex electromagnetic environment Compared to the one signal method the dichotomous method allows one to carry out a much more faster search by factor of tens or more An example of the search process and an estimation of the number of the required analysis cycles as well as description of the algorithm are given The relation between identification and optimization problems is outlined INTRODUCTION Computer aided modeling of a radio electronic system is a very useful tool for electromagnetic compatibility interference EMC EMI analysis in that it allows for the simulation of system behavior for a wide variety of initial conditions excitations and system configurations in a rapid and inexpensive way 1 A system can often reveal nonlinear behavior and nonlinear phenomena intermodulation cross modulation gain compression expansion etc has profound effect on EMC EMI in some cases 2 Taking into account nonlinear interference at the system subsystem design phase makes it possible to reduce th
4. al Engineering Section Mc Donnel Douglas Aircraft Corp F30602 72 C 0277 Rome Air Development Center Griffiss AFB NY Dec 1974 10 F M Tesche A W Kalin M Nyffeler and B R Brandli Representation of Wide Band Spectra Using An Adaptive Nonuniform Sampling Scheme Proc of 12 Inter Zurich Symp EMC Zurich Feb 18 20 1997 11 W H Press B P Flanneay S A Teukolsky W T Vetterling Numerical Recipes in C Cambridge University Press 1988 12 1D Cheremisinov S L Loyka V I Mordachev Synthesis of the polynomial model of nonlinear elements based on intermodulation dynamic ranges Proc of 3 Inter Confer On Telecommunications in Modern Satellite Cable and Broadcasting Services TELSIKS 97 Oct 8 10 Nis Yugoslavia 1997 pp 519 522 13 S L Loyka Detector Simulation with the Use of the Discrete Technique 14th Inter Wroclaw Symp on EMC Poland Wroclaw to be published 14 Yu I Degtjarev Optimization Sovetskoe Radio 1980 In Russian Methods Moscow 15 M Aoki Introduction to Optimization Techniques The Macmillan Company New York 1976
5. e cost of its removal considerably An identification of nonlinear interference sources is a very important task from the viewpoint of their removal A computer aided simulation tool can be used for such an identification in a very efficient way 3 This article deals with a method of automatic identification of nonlinear interference sources which is used in order to solve EMC EMI problems in complex electromagnetic environment for instance in mobile communications environment where there is a lot of emitters and receptors of EMI The specific character of this task is that a very accurate simulation of signals and interference levels is not required However in this case the analysis of complex systems must be carried out Because of this the simulation should be carried out at the system level A nonlinear modeling technique so called discrete technique for numerical EMC EMI simulation at the system level has been proposed in 4 5 This technique allows one to carry out rapid numerical EMC EMI analysis of a complex system or subsystem i e receiver transmitter etc or a set of systems subsystems in a wide frequency range taking into Vladimir Mordachev Belorussian State University of Informatics amp Radioelectronics Brovki Str 6 Minsk 220027 BELARUS account nonlinear effects including spurious responses of a receiver and maintaining accurate spectra representation Such an analysis is for instance a very
6. f nonlinear interference sources by the dichotomous method The total number of signals N 8 S and S are the sources of the nonlinear interference The number of analysis cycles required to identify an interference which has k sources is ny 2k log N 9 The comparison of 8 with 9 shows that the dichotomous search method provides considerable advantage over the one signal method when there is a large number of signals Here is an example For k 2 and N 0 the one signal method requires for 1000 analysis cycles and the dichotomous method for about 40 analysis cycles for higher N values this difference is even more pronounced If one analysis cycle takes 1 minute to carry out then the analysis with the use of the one signal method will last for about 16 hours and the analysis with the use of the dichotomous method will last for about 40 minutes this difference is similar to the difference between discrete Fourier transform and fast Fourier transform The search time can be significantly reduced if the signals are previously sorted in accordance with their amplitude and the group which contains the smaller signals is excluded from the analysis in the first turn since large signals are the most probable nonlinear interference sources It is expedient to take into consideration the intermodulation dynamic range of the receiver It is also expedient to determine whether the signals fall into the RF preselector bandwidth the signa
7. her orders which may be formed by more than 2 signals and for the whole class of other nonlinear interference types desensitization cross modulation local oscillator noise conversion etc This principle may be used as a basis for a number of identification methods which consist in repeated recalculation of the signal at the receiver output while one or several sources are excluded from the analysis The simplest identification method is 1 to carry out the calculation of the output signal when all the signals S Sy are active turned on 2 to exclude to turn off the signal S 3 make the analysis i e computation of the total signal at the receiver output for the other signals S2 Sy 4 check whether the interference disappeared The interference amplitude A is an indicator of the disappearance Ain lt OA into gt 7 where Ainso is interference level at the step 1 when the signal S was turned on is a reduction in the interference level which indicates its disappearance O 0 5 0 1 If the interference did not disappear then S is not its source otherwise it is its source 5 Then the procedure is repeated for the signals S Sy This method may be called the one signal method Its use is expedient when the signals number N is not large lt 10 since the nonlinear receiver analysis itself requires for a lot of time this value may vary from several seconds up to severa
8. hich describes the transfer characteristic of the nonlinear element J is order of the polynomial The necessity of polynomial approximation of the nonlinear element transfer characteristic will be substantiated below The transition from the time domain to the frequency domain and vise versa is made with the use of the direct and inverse fast Fourier Transform FFT The direct FFT can be carried out by one of known methods 11 using the following ratio s i Sn S fa 5 SAP up ult ult Af frequency sample interval At time sample interval N number of samples The inverse FFT is WwW g AON 3 where N 1 m S W 4 n 0 It is worth mentioning that the normalization given in 3 and 4 must be used during a nonlinear analysis The normalization of other types which is often used in the literature will produce incorrect results The direct and inverse FFT vary only in the normalization and the exponent sign which makes it possible to use the same algorithm in order to carry out the direct as well as the inverse FFT It is necessary to make the corresponding data normalization and to arrange the data in the appropriate order before the FFT is carried out Let us note a number of peculiarities connected with the use of the FFT for nonlinear analysis 1 The maximum frequency in the spectrum Fmax frequency sample interval Af time sample interval At and the number of samples N are connected by the foll
9. important part of EMC EMI modeling of a mobile communication system 6 8 THE DISCRETE TECHNIQUE The basis of the discrete technique 4 5 is a representation of the equivalent block diagram of a system as linear filters LF and memoryless nonlinear elements MNE connected in series or in parallel Thus a stage which employs a nonlinear element for example an amplifier can be represented as a typical radio stage see Figure 1 which employs the linear filter at the input the memoryless nonlinear element and the linear filter at the output 2 A Linear Filter Input Memoryless Nonlinear Element Linear Filter Output Figure 1 Representation of a typical radio frequency stage This representation reflects characteristic peculiarities inherent to the construction of typical amplifying and converting stages The utilization of the model with memoryless nonlinearity is not a significant limitation on the method for two reasons First non zero memory effects can partially be factorized at the level of input or output filters that is this representation is equivalent with respect to the simulation of the input to output link Second the prediction of a signal spectrum at the system input taking into consideration EMC problems is as a tule not very accurate the error can be as large as several dB or even tens of dB It is an essential limitation on the simulation accuracy the accuracy a signal at the system
10. l hours depending on the receiver complexity and a computer type The required number of analysis cycles is This method cannot be used if there is a large number of signals In this case it is necessary to use the dichotomous search method DICHOTOMOUS SEARCH METHOD The essence of this method is as follows a group of signals rather than each separate signal is turned off If the exclusion of the group of signals does not cause the interference to disappear then this group of signals does not contain interference sources and can be discarded from the further consideration If the interference does disappear then this group contains an interference source In this case the group is to be divided into parts and these parts are to be analyzed with the use of the method described above When the dichotomous method is used the group under analysis is divided into 2 equal parts at each step This process is repeated until each group contains one signal whose exclusion makes it possible to determine whether or not this signal is an interference source This method is schematically represented in Figure 3 In the case under consideration there are 8 signals S Sg the signals S and S are the interference sources Each group of signals is divided into two parts at each step of the analysis The parts whose exclusion does not cause the interference to disappear are discarded from the further search steps Figure 3 An example of the search o
11. ls which do not fall into the RF preselector bandwidth are excluded in the first turn Start A Porting signals according to the increase in power V The identified interference is chosen Vv CSS All Signals L 1 CSS is divided into 2 parts Vv Turning off CSS Vv Analysis V Did interference disappear Yes gt No Y Excluding CSS from CSS CSS CSS CSS 4 Is only one signal left in CSS Yes The signal is recorded to the interference file 6 66 Figure 4 The Dichotomous Search Algorithm CSS current sets of signals MNIS the maximum number of interference signals L its current value i internal variable 0 In CSS one signal Yes The signal is recorded to the interference file a NSS CSS 1 lt V In CSS one signal Yes Y The signal is recorded to the interference file Vv M 0 Y Yes Return to the upper nesting level V Identification of next i 9 gt interference Yes No End Figure 4 The Dichotomous Search Algorithm continued NSS new sets of
12. nd the order of nonlinearity are specified 3 The maximum possible range of amplitudes in the signal spectrum is determined by errors in the signal amplitude quantization in the time domain that is by the accuracy of computer data presentation for a floating point number with double format this value is 280 dB When simulating multistage systems the quantization noise caused by the amplitude quantization is accumulated This effect can be nullified by periodic clearing of the spectrum that is zeroing of the components whose level is lower than a certain threshold 4 The utilization of geometrically spaced sample frequencies makes it possible to reduce the number of samples or to reduce the frequency sample interval or to increase the order of simulated nonlinearity However it will slightly increase the simulation time Further improvement in the computational efficiency of the radio systems simulation can be achieved by means of a two stage simulation scheme 4 At the first stage the radio system simulation correct to carrier frequencies low frequency resolution is carried out All interference signals revealed at the first stage are sequentially analyzed at high frequency resolution correct to modulating spectra and with transformation to low frequencies A polynomial synthesis technique has been discussed in 12 A detector can also be simulated by means of this technique 13 Using the technique a radio recei
13. output can be predicted with Thus great accuracy of system simulation is not necessarily required when the input signal is known with small accuracy Therefore our viewpoint is that the utilization of the Volterra series for the analysis of nonlinear effects with respect to EMC problems 2 causes an essential increase of complexity without any essential increase in the analysis accuracy taken as a whole The process of signal passage through linear filters is simulated in the frequency domain using the complex transfer factor of the filter Soul Pa Snl GOR sale 1 where Sou fn is the signal spectrum at the filter output S f the signal spectrum at the filter input K f is the complex transfer factor of the filter fp are sample frequencies It is necessary to have a sampled spectrum in order to do a calculation of this type A spectrum sampling technique is given in 9 The essential improvements in this technique which adopt it for modern computers and allow one to increase accuracy are given in 3 The adaptive sampling technique can be used for this purpose too 10 The process of signal passage through a nonlinear memoryless element is simulated in the time domain I Yau ty 2 i l You t where u t is the instantaneous value of the signal at the MNE output w t is the same for the MNE input t are sample points in time a are coefficients of the high order polynomial w
14. owing ratios T 1 At N 5 pice ieee At At Af where 7 Af signal repetition period The necessary number of samples in the frequency domain is actually equal to N 2 since samples with numbers arranged symmetrically with respect to N 2 are complex conjugate ones Sy S In the time domain all N samples are independent 2 Nonlinear transformation of the input signal causes its spectrum to expand times power of the polynomial which describes the amplitude characteristic of the nonlinear element therefore taking into account the cyclic character of the FFT in the frequency domain 11 the maximum allowable frequency in the input signal spectrum will be 2 F 1 Fn max a gt gt 6 i I 1 I 1 At Thus the undistorted spectrum is obtained at the nonlinear element output within the interval 0 Finmax Hence it is clear why the polynomial approximation 2 is to be used for the nonlinear element characteristic otherwise the spectrum would expand infinitely which would produce incorrect results When the inverse FFT is calculated at the nonlinear element output the spectrum S has to be calculated only within the interval 0 Finmax which allows one to reduce the calculation time The ratios 5 6 make it possible to determine the number of samples and hence the amount of computer memory which is required in order to analyze a system if the maximum frequency at the input frequency sample interval a
15. s L exceeds the maximum admissible value MNIS which is set by the user If it does not then the search procedure is continued if it does then the process of the current interference sources search is stopped and the user can choose a next interference to search its sources the limitation of the interference sources number is necessary in order to limit the time the search process requires and the number of sequential calls to the nested procedure DIVIDE TWO If there is only one signal in each part then they are interference sources and then the exit from the procedure is made Otherwise the parts which contain more than one signal are divided into two parts and the above mentioned operations are repeated two new sets of signals are introduced and the procedure DIVIDE _TWO is called again After the exit from the procedure DIVIDE _TWO of the uppermost level the user can choose a next interference for identifying If it is not necessary then the algorithm is completed The interference signals have been saved to the interference file THE RELATION BETWEEN IDENTIFICATION AND OPTIMIZATION PROBLEMS It should be pointed out that the problem of the interference source identification is similar to some optimization problems 14 15 If we define the goal function F as a function of several signals each interference has its own goal function F F Snp Sn2 see Sn 10 where k is the number of interference sources in such a way tha
16. t this function is equal to 1 for the interference sources and to 0 for all other combinations of signals r if Sni Snk is the full set of interference sources 0 otherwise 1 then the identification problem will be completely similar to the problem of maximizing F which can be solved with the use of a number of well known techniques 14 15 among which are the Fibonacci method the golden section method as well as the dichotomous method CONCLUSIONS The computer aided method of nonlinear interference sources identification has been presented in this paper This method can be applied for the identification of nonlinear interference intermodulation cross modulation desensitization etc sources in a complicated electromagnetic environment when there is a lot of interference for instance in mobile communications in co site situations etc and when it s difficult to find out interference sources manually Further improvement in the computational efficiency of the identification technique can be achieved by use of methods known from optimization theory The technique proposed can also be used for the identification of linear interference sources However this is unsuitable because identification of this interference can be carried out at the stage of linear analysis which requires less time REFERENCES 1 F M Tesche Numerical Modeling for EMC Proc of 12 Inter Zurich Symp On EMC Zurich Switzerland Feb 18 20
17. ver can be simulated in a wide frequency range with very high frequency resolution up to 10 10 sample frequencies on a modern PC in dozens of minutes a conventional circuit level simulation would require several years for such an analysis IDENTIFICATION OF NONLINEAR INTERFERENCE SOURCES Next we will consider the simulation of radio receivers all obtained results can be easily applied to systems of other kinds too A situation under analysis is shown in Figure 2 Interference signals S Sy separate spectral components of signals can also be used as S affect the victim receiver Rx and cause nonlinear interference at its output Figure 2 Situation under analysis Interference signals S Sy affect the victim receiver Rx and cause nonlinear interference at its output In the general case the problem of nonlinear interference sources identification is much more complex than the interference sources identification during linear analysis The general approach to nonlinear interference sources identification may be formulated on the basis of the fact that a nonlinear interference disappears when at least one signal which takes part in its formation is excluded is turned off For example a second order intermodulation product is proportional to the product of amplitudes of signals which take part in its formation IMP U U If U 0 then IMP 0 the same for U2 A similar principle is also true for the case of IMP of hig
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