November 2005
doc.: IEEE 802.19-05/0043r0
A Method of Curve Fitting to BER Data
Date: 2005-11-01
Authors:
Name
Company Address
Phone
E-mail
Steve
Shellhammer
Qualcomm
(858) 658-1874
[email protected]
5775 Morehouse Dr
San Diego, CA 92121
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Submission
Slide 1
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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
Submission
Slide 2
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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
Slide 3
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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)
f ( )
• 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
1
f (0) 
2
Submission
Lim f ( )  0
 
Slide 4
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
Functional Curve Fitting
• Given the exponential nature of typical BER curves the
following functional format is proposed
f ( ) 
1
exp( g ( ))
2
• Where the function g is a polynomial with no constant
term
• An example of this is,
1
f ( )  exp( a  b 2 )
2
Submission
Slide 5
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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,
{ i , pi }
• 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
Submission
Slide 6
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
Functional Curve Fitting
• The BER formula says,
1
p  f ( )  exp( a  b 2 )
2
• Multiplying both sides by two and taking natural
logarithms gives,
Log (2 p)  a  b 2
Submission
Slide 7
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
Functional Curve Fitting
• Applying the N BER data measurement pairs to this
equations gives the following N linear equations
Log (2 pi )  a i  b i
2
• 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
Slide 8
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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
Submission
SNR
BER
0.0
0.080400000000
1.0
0.061800000000
2.0
0.035000000000
3.0
0.024600000000
4.0
0.013950000000
5.0
0.005650000000
6.0
0.002344444444
7.0
0.000762962963
8.0
0.000187962963
9.0
0.000033612040
10.0
0.000004100000
Slide 9
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
Example – BPSK Simulation
• Results of least squared solution for coefficients a and b
1
f ( )  exp( 1.5136   0.036265  2 )
2
• 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
Submission
Slide 10
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
Example – BPSK Simulation
Submission
Slide 11
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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
Submission
Slide 12
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
Example – 802.15.4b PSSS in 915 MHz
Submission
Slide 13
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
Example – 802.15.4b PSSS in 868 MHz
Submission
Slide 14
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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
Submission
Slide 15
Steve Shellhammer, Qualcomm Inc.
November 2005
doc.: IEEE 802.19-05/0043r0
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
Slide 16
Steve Shellhammer, Qualcomm Inc.