1. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 1 A Method of Curve Fitting to BER Data 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 IEEEs name any IEEE Standards publication even though it may include portions of this contribution; and at the IEEEs 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, 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 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.http:// ieee802.org/guides/bylaws/sb-bylaws.pdfpatcom@ieee.org Date: 2005-11-01 Authors:

2. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 2 Presentation Outline Motivation Functional Curve Fitting Examples –BPSK Data –802.15.4b PSSS in 915 MHz (US) –802.15.4b PSSS in 868 MHz (Europe) Word document IEEE 802.19-05/0042r0

3. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 3 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

4. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 4 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 dB) 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

5. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 5 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,

6. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 6 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 dB into a linear scale

7. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 7 Functional Curve Fitting Multiplying both sides by two and taking natural logarithms gives, The BER formula says,

8. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 8 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

9. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 9 Example – BPSK Simulation The first example is based on a simple BPSK simulation The following simulation data was used The SNR was in dB and needed to be converted to a linear scale SNRBER 0.00.080400000000 1.00.061800000000 2.00.035000000000 3.00.024600000000 4.00.013950000000 5.00.005650000000 6.00.002344444444 7.00.000762962963 8.00.000187962963 9.00.000033612040 10.00.000004100000

10. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 10 Example – BPSK Simulation Since the coefficient b is positive this function only work for up to around 20 dB. After that point you need to set the BER to zero, which is a very good approximation Results of least squared solution for coefficients a and b

11. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 11 Example – BPSK Simulation

12. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 12 Example – 802.15.4b PSSS Data for the 802.15.4b 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

13. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 13 Example – 802.15.4b PSSS in 915 MHz

14. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 14 Example – 802.15.4b PSSS in 868 MHz

15. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 15 Example – 802.15.4b 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

16. doc.: IEEE 802.19-05/0043r0 Submission November 2005 Steve Shellhammer, Qualcomm Inc.Slide 16 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