berfit - Fit curve to nonsmooth empirical bit error rate (BER) data

Syntax

fitber = berfit(empEbNo,empber) fitber = berfit(empEbNo,empber,fitEbNo) fitber = berfit(empEbNo,empber,fitEbNo,options) fitber = berfit(empEbNo,empber,fitEbNo,options,fittype) [fitber,fitprops] = berfit(...) berfit(...) berfit(empEbNo,empber,fitEbNo,options,'all')

Description

fitber = berfit(empEbNo,empber) fits a curve to the empirical BER data in the vector empber and returns a vector of fitted bit error rate (BER) points. The values in empber and fitber correspond to the E_b/N₀ values, in dB, given by empEbNo. The vector empEbNo must be in ascending order and must have at least four elements.

Note The berfit function is intended for curve fitting or interpolation, not extrapolation. Extrapolating BER data beyond an order of magnitude below the smallest empirical BER value is inherently unreliable.

fitber = berfit(empEbNo,empber,fitEbNo) fits a curve to the empirical BER data in the vector empber corresponding to the E_b/N₀ values, in dB, given by empEbNo. The function then evaluates the curve at the E_b/N₀ values, in dB, given by fitEbNo and returns the fitted BER points. The length of fitEbNo must equal or exceed that of empEbNo.

fitber = berfit(empEbNo,empber,fitEbNo,options) uses the structure options to override the default options used for optimization. These options are the ones used by the fminsearch function. You can create the options structure using the optimset function. Particularly relevant fields are described in the table below.

Field	Description
`options.Display`	Level of display: `'off'` (default) displays no output; `'iter'` displays output at each iteration; `'final'` displays only the final output; `'notify'` displays output only if the function does not converge.
`options.MaxFunEvals`	Maximum number of function evaluations before optimization ceases. The default is 10⁴.
`options.MaxIter`	Maximum number of iterations before optimization ceases. The default is 10⁴.
`options.TolFun`	Termination tolerance on the closed-form function used to generate the fit. The default is 10^-4.
`options.TolX`	Termination tolerance on the coefficient values of the closed-form function used to generate the fit. The default is 10^-4.

fitber = berfit(empEbNo,empber,fitEbNo,options,fittype) specifies which closed-form function berfit uses to fit the empirical data, from the possible fits listed in Algorithm below. fittype can be 'exp', 'exp+const', 'polyRatio', or 'doubleExp+const'. To avoid overriding default optimization options, use options = [].

[fitber,fitprops] = berfit(...) returns the MATLAB structure fitprops, which describes the results of the curve fit. Its fields are described in the table below.

Field	Description
`fitprops.fitType`	The closed-form function type used to generate the fit: `'exp'`, `'exp+const'`, `'polyRatio'`, or `'doubleExp+const'`.
`fitprops.coeffs`	The coefficients used to generate the fit. If the function cannot find a valid fit, `fitprops.coeffs` is an empty vector.
`fitprops.sumSqErr`	The sum squared error between the log of the fitted BER points and the log of the empirical BER points.
`fitprops.exitState`	The exit condition of `berfit`: `'The curve fit converged to a solution.'`, `'The maximum number of function evaluations was exceeded.'`, or `'No desirable fit was found'`.
`fitprops.funcCount`	The number of function evaluations used in minimizing the sum squared error function.
`fitprops.iterations`	The number of iterations taken in minimizing the sum squared error function. This is not necessarily equal to the number of function evaluations.

berfit(...) plots the empirical and fitted BER data.

berfit(empEbNo,empber,fitEbNo,options,'all') plots the empirical and fitted BER data from all the possible fits, listed in the Algorithm below, that return a valid fit. To avoid overriding default options, use options = [].

Note A valid fit must be

real-valued
monotonically decreasing
greater than or equal to 0 and less than or equal to 0.5

If a fit does not confirm to this criteria, it is rejected.

Algorithm

The berfit function fits the BER data using unconstrained nonlinear optimization via the fminsearch function. The closed-form functions that berfit considers are listed in the table below, where x is the E_b/N₀ in linear terms (not dB) and f is the estimated BER. These functions were empirically found to provide close fits in a wide variety of situations, including exponentially decaying BERs, linearly varying BERs, and BER curves with error rate floors.

Value of `fittype`	Functional Expression
`'exp'`
`'exp+const'`
`'polyRatio'`
`'doubleExp+const'`

The sum squared error function that fminsearch attempts to minimize is

where the fitted BER points are the values in fitber and the sum is over the E_b/N₀ points given in empEbNo. It is important to use the log of the BER values rather than the BER values themselves so that the high-BER regions do not dominate the objective function inappropriately.

Examples

The examples below illustrate the syntax of the function, but they use hard-coded or theoretical BER data for simplicity. For an example that uses empirical BER data from a simulation, see Example: Curve Fitting for an Error Rate Plot.

The code below plots the best fit for a sample set of data.

EbNo = 0:13;
berdata = [.2 .15 .13 .12 .08 .09 .08 .07 .06 .04 .03 .02 .01 .004];
berfit(EbNo,berdata); % Plot the best fit.

The curve connects the points created by evaluating the fit expression at the values in EbNo. To make the curve look smoother, use a syntax like berfit(EbNo,berdata,[0:0.2:13]). This alternative syntax uses more points when plotting the curve, but it does not change the fit expression.

The next example demonstrates a fit for a BER curve with an error floor. We generate the empirical BER array by simulating a channel with a null (ch = [0.5 0.47]) with BPSK modulation and linear MMSE equalizer at the receiver. We run the berfit with the 'all' option. The 'doubleExp+const' fit does not provide a valid fit, and the 'exp' fit type does not work well for this data. The 'exp+const' and 'polyRatio' fits closely match the simulated data.

EbNo = -10:3:15;
empBER = [0.3361 0.3076 0.2470 0.1878 0.1212 0.0845 0.0650 0.0540 0.0474];
figure; berfit(EbNo, empBER, [], [], 'all');

The following code illustrates the use of the options input structure as well as the fitprops output structure. The 'notify' value for the display level causes the function to produce output when one of the attempted fits does not converge. The exitState field of the output structure also indicates which fit converges and which fit does not.

M = 4; EbNo = 3:10;
berdata = berfading(EbNo,'psk',M,2); % Compute theoretical BER.
noisydata = berdata.*[.93 .92 1 .59 .08 .15 .01 .01];
% Say when fit fails to converge.
options = optimset('display','notify');

disp('*** Trying exponential fit.') % Poor fit
[fitber1,fitprops1] = berfit(EbNo,noisydata,EbNo,...
   options,'exp')

disp('*** Trying polynomial ratio fit.') % Good fit
[fitber2,fitprops2] = berfit(EbNo,noisydata,EbNo,...
   options,'polyRatio')

The output is as follows:

*** Trying exponential fit.
 
Exiting: Maximum number of function evaluations has been exceeded
         - increase MaxFunEvals option.
         Current function value: 2.729948 


fitber1 =

    0.0766    0.0423    0.0205    0.0086    0.0030    0.0009    0.0002    0.0000


fitprops1 = 

       fitType: 'exp'
        coeffs: [4x1 double]
      sumSqErr: 2.7299
     exitState: 'The maximum number of function evaluations has been exceeded'
     funcCount: 10000
    iterations: 6177

*** Trying polynomial ratio fit.

fitber2 =

    0.0931    0.0476    0.0220    0.0090    0.0031    0.0008    0.0001    0.0001


fitprops2 = 

       fitType: 'polyRatio'
        coeffs: [6x1 double]
      sumSqErr: 2.0578
     exitState: 'The curve fit converged to a solution'
     funcCount: 580
    iterations: 344

References

For a general description of unconstrained nonlinear optimization, see the following work.

[1] Chapra, Steven C., and Raymond P. Canale, Numerical Methods for Engineers, Fourth Edition, New York, McGraw-Hill, 2002.

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