November 2005 doc IEEE 802 19 050043 r

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 A Method of Curve Fitting to BER Data Authors: Date: 2005 -11 -01 Notice: This document has been prepared to assist IEEE 802. 19. It is offered as a basis for discussion and is not binding on the contributing individual(s) or organization(s). The material in this document is subject to change in form and content after further study. The contributor(s) reserve(s) the right to add, amend or withdraw material contained herein. Release: The contributor grants a free, irrevocable license to the IEEE to incorporate material contained in this contribution, and any modifications thereof, in the creation of an IEEE Standards publication; to copyright in the IEEE’s name any IEEE Standards publication even though it may include portions of this contribution; and at the IEEE’s sole discretion to permit others to reproduce in whole or in part the resulting IEEE Standards publication. The contributor also acknowledges and accepts that this contribution may be made public by IEEE 802. 19. Patent Policy and Procedures: The contributor is familiar with the IEEE 802 Patent Policy and Procedures <http: // ieee 802. org/guides/bylaws/sb-bylaws. pdf>, including the statement "IEEE standards may include the known use of patent(s), including patent applications, provided the IEEE receives assurance from the patent holder or applicant with respect to patents essential for compliance with both mandatory and optional portions of the standard. " Early disclosure to the TAG of patent information that might be relevant to the standard is essential to reduce the possibility for delays in the development process and increase the likelihood that the draft publication will be approved for publication. Please notify the Chair <shellhammer@ieee. org> as early as possible, in written or electronic form, if patented technology (or technology under patent application) might be incorporated into a draft standard being developed within the IEEE 802. 19 TAG. If you have questions, contact the IEEE Patent Committee Administrator at <patcom@ieee. org>. Submission 1 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Presentation Outline • Motivation • Functional Curve Fitting • Examples – BPSK Data – 802. 15. 4 b PSSS in 915 MHz (US) – 802. 15. 4 b PSSS in 868 MHz (Europe) • Word document IEEE 802. 19 -05/0042 r 0 Submission 2 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Motivation • Estimating the packet error rate (PER) caused by interference requires the BER or possibly the symbol error rate (SER) • Many times the BER curves are developed using simulations since there is no analytic expression for the BER • Though it is possible to use the tabulated BER data in estimating the PER it is often more convenient to utilize a formula • Also the PER calculations are likely to required BER values outside the range of the original BER simulation data, so somehow that BER data needs to be extrapolated Submission 3 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Functional Curve Fitting • The approach suggested here is to select a parameterized function and select the parameters of the function to fit the BER simulation data • The BER is a function of the signal to noise ratio (SNR) • The SNR is on a linear (not d. B) scale • Based on principals of probability, the function needs to meet two boundary conditions – At zero SNR the BER must be one-half – At infinite SNR the BER must be zero Submission 4 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Functional Curve Fitting • Given the exponential nature of typical BER curves the following functional format is proposed • Where the function g is a polynomial with no constant term • An example of this is, Submission 5 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Functional Curve Fitting • This format guarantees that for zero SNR that the BER is one-half • If the coefficient b is negative then the BER tends to zero as the SNR goes to infinity • The simulation data typically consists of a sequence of pair of the format, • The SNR is assumed in this analysis to be in a linear scale. If that is not the case the first step is to convert the SNR from d. B into a linear scale Submission 6 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Functional Curve Fitting • The BER formula says, • Multiplying both sides by two and taking natural logarithms gives, Submission 7 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Functional Curve Fitting • Applying the N BER data measurement pairs to this equations gives the following N linear equations • The final step in the process is to find the least squares estimate for the two unknowns (a and b) given these N linear equations Submission 8 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Example – BPSK Simulation • The first example is based on a simple BPSK simulation • The following simulation data was used • The SNR was in d. B and needed to be converted to a linear scale Submission SNR 9 BER 0. 08040000 1. 0 0. 06180000 2. 0 0. 03500000 3. 0 0. 02460000 4. 0 0. 013950000000 5. 0 0. 005650000000 6. 0 0. 00234444 7. 0 0. 000762962963 8. 0 0. 000187962963 9. 0 0. 000033612040 10. 000004100000 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Example – BPSK Simulation • Results of least squared solution for coefficients a and b • Since the coefficient b is positive this function only work for up to around 20 d. B. • After that point you need to set the BER to zero, which is a very good approximation Submission 10 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Example – BPSK Simulation Submission 11 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Example – 802. 15. 4 b PSSS • Data for the 802. 15. 4 b parallel sequence spread spectrum (PSSS) was supplied by Andreas Wolf • The presentation will show the results of the curve fitting • More detail is available in the Word document Submission 12 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Example – 802. 15. 4 b PSSS in 915 MHz Submission 13 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Example – 802. 15. 4 b PSSS in 868 MHz Submission 14 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Example – 802. 15. 4 b PSSS in 868 MHz • One observation about the results of curve fitting to the 868 MHz data is that the fit is not that good a low SNR • The reason for this is the simulation results show that at low SNR the BER appears to be approaching a value less than one-half. The reason for this is unknown Submission 15 Steve Shellhammer, Qualcomm Inc.

November 2005 doc. : IEEE 802. 19 -05/0043 r 0 Conclusions • A method of fitting a functional curve to a set of BER simulation results was presented • The functional format is exponential in nature as a typical BER curve • The format is such that the BER is guaranteed to be one-half at low SNR • As long at the highest order coefficient turns out negative the BER tend to zero at high SNR • If the highest order coefficient is positive then you need to use this approximation up to some specified SNR and set the BER to zero for higher SNR • It might be possible to get a better fit if we fit functions to sections of the BER data and end up with a piecewise functional format Submission 16 Steve Shellhammer, Qualcomm Inc.