berfit

Fit curve to nonsmooth empirical 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, empber, and returns a vector of fitted BER points. The values in empber and fitber correspond to the empirical energy per bit to noise power spectral density ratio (E_b/N₀) values given by empEbNo. For a general description of unconstrained nonlinear optimization, see [1].

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) specifies a vector of E_b/N₀ values to use when fitting a curve to the empirical BER data in empber that correspond to the empirical E_b/N₀ values in empEbNo.

fitber = berfit(empEbNo,empber,fitEbNo,options) specifies a structure to override the default options used for optimization.

fitber = berfit(empEbNo,empber,fitEbNo,options,fittype) specifies the closed-form function used to fit the empirical data. If you do not want to override the default options for optimization, specify options as [ ].

[fitber,fitprops] = berfit(___) returns the fitprops structure with fields that describe the properties of the curve fit. Use any input argument combination from previous syntaxes.

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

berfit(empEbNo,empber,fitEbNo,options,'all') plots the empirical and fitted BER data from all possible settings of fittype that are valid. If you do not want to override the default options for optimization, specify options as [ ].

Note

To be valid a fit must conform to these criteria, otherwise it is rejected.

• real-valued

• monotonically decreasing

• greater than or equal to 0 and less than or equal to 1

Examples

Fit Curve to BER Points

This example shows the use of the berfit function using hard-coded or theoretical BER points for simplicity. For an example that uses empirical BER data from a simulation, see Use Curve Fitting on Error Rate Plot.

Best Fit for Set of Sample Data

Define a range of ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ values and BER points. Use this data as inputs for the berfit function.

EbN0 = 0:13;
berdata = [.2 .15 .13 .12 .08 .09 .08 .07 .06 .04 .03 .02 .01 .004];
berfit(EbN0,berdata); 

Plot Best Fit

The curve connects the points created by evaluating the fit expression at the specified ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ values. To make the curve look smoother, provide an input vector of ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ values for curve fitting in ascending order. This vector provides more points for plotting the curve and does not change the fit expression.

fitEbNo = 0:0.2:13;
berfit(EbN0,berdata,fitEbNo)

Fit for BER Curve with Error Floor

Run the berfit function using the 'all' option on empirical BER results for a simulation of BPSK data transmitted over a channel with a null (ch = [0.5 0.47]) and recovered by using a linear MMSE equalizer at the receiver for the ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ range [-10, 15]. Comparing the results of curve fitting methods

• 'doubleExp+const' fit type does not provide a valid fit

• 'exp' fit type does not work well for this data

• 'exp+const' and 'polyRatio' fit types closely match the simulated data

EbN0 = -10:3:15;
empBER = [0.3361 0.3076 0.2470 0.1878 0.1212 0.0845 0.0650 0.0540 0.0474];
figure;
berfit(EbN0,empBER,[],[],'all');

Use of options Input Structure and 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 indicates which fit type converges.

Generate theoretical BER results for 8-PSK data with a diversity order of 2 transmitted over a Rayleigh fading channel for the ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ range [3, 10] dB.

M = 8; EbN0 = 3:10;
berdata = berfading(EbN0,'psk',M,2); % Compute the theoretical BER
noisydata = berdata.*[.93 .92 1 .59 .08 .15 .01 .01];

Create the options structure by using the optimset function to configure display and notification of fit type results. Run exponential and polynomial ratio fit types.

options = optimset('display','notify');
disp('*** Trying exponential fit.') % Poor fit

*** Trying exponential fit.

[fitber1,fitprops1] = berfit(EbN0,noisydata,EbN0,...
   options,'exp')

 
Exiting: Maximum number of function evaluations has been exceeded
         - increase MaxFunEvals option.
         Current function value: 2.749919 

fitber1 = 1×8

    0.1247    0.0727    0.0376    0.0168    0.0064    0.0020    0.0005    0.0001

fitprops1 = struct with fields:
       fitType: 'exp'
        coeffs: [4x1 double]
      sumSqErr: 2.7499
     exitState: 'The maximum number of function evaluations has been exceeded'
     funcCount: 10001
    iterations: 6193

disp('*** Trying polynomial ratio fit.') % Good fit

*** Trying polynomial ratio fit.

[fitber2,fitprops2] = berfit(EbN0,noisydata,EbN0,...
   options,'polyRatio')

fitber2 = 1×8

    0.1701    0.0874    0.0407    0.0169    0.0060    0.0016    0.0003    0.0001

fitprops2 = struct with fields:
       fitType: 'polyRatio'
        coeffs: [6x1 double]
      sumSqErr: 2.3880
     exitState: 'The curve fit converged to a solution'
     funcCount: 554
    iterations: 331

Input Arguments

empEbNo — Empirical E_b/N₀ values

vector

Empirical E_b/N₀ values in dB, specified as a vector with at least four elements. The element values in the vector must be in ascending order.

Data Types: single | double

empber — Empirical BER data

vector

Empirical BER data, specified as a vector with the same number of elements as input empEbNo. The values in empber correspond to the E_b/N₀ values given by empEbNo.

Data Types: single | double

fitEbNo — E_b/N₀ values for curve fitting

vector

E_b/N₀ values in dB for curve fitting, specified as a vector with element values in ascending order. The length of fitEbNo must equal or exceed that of input empEbNo.

Data Types: single | double

options — Override default options used for optimization

[ ] | structure

Override default options used for optimization, specified as a structure. The fields specified in the options structure are used by the fminsearch function. You can create the options structure by using the optimset function. This table describes the fields that are most relevant when using the berfit function. To use default options, you can specify this input as [ ].

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	The maximum number of function evaluations before optimization ceases. The default is 104.
options.MaxIter	The maximum number of iterations before optimization ceases. The default is 104.
options.TolFun	The termination tolerance for the closed-form function used to generate the fit. The default is 10-4.
options.TolX	The termination tolerance for the coefficient values of the closed-form function used to generate the fit. The default is 10-4.

Data Types: struct

fittype — Closed-form function used to fit empirical data

'exp' | 'exp+const' | 'polyRatio' | 'doubleExp+const'

Closed-form function used to fit the empirical data, specified as 'exp', 'exp+const', 'polyRatio', or 'doubleExp+const'. For more information, see Algorithms.

Data Types: char | string

Output Arguments

fitber — Fitted BER points

vector

Fitted BER points, returned as a vector. The BER is computed for each E_b/N₀ setting specified by the input empEbNo vector.

Data Types: double

fitprops — Fit properties

structure

Fit properties, returned as a structure with these fields to describe the properties of the curve fit.

Field	Description
fitprops.fitType	The closed-form function type used to generate the fit. Valid values include: '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. Valid values include: 'The curve fit converged to a solution.' 'The maximum number of function evaluations was exceeded.' '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 value is not necessarily equal to the number of function evaluations.

Data Types: struct

Algorithms

The berfit function fits the BER data using unconstrained nonlinear optimization via the fminsearch function. This table lists the closed-form functions that berfit considers based on the value of the fittype input argument. 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. In the functional expressions, x is a linear E_b/N₀ value (not a dB value), and f(x) is the estimated BER.

fittype Value	Functional Expression
'exp'	$$f(x)=a_1\exp \left[\frac{-{\left(x-a_2\right)}^{a_3}}{a_4}\right]$$
'exp+const'	$$f(x)=a_1\exp \left[\frac{-{(x-a_2)}^{a_3}}{a_4}\right]+a_5$$
'polyRatio'	$$f(x)=\frac{a_1{x}^2+a_{2}x+a_3}{x^3+a_4{x}^2+a_{5}x+a_6}$$
'doubleExp+const'	$$\begin{array}{l}{a}_1\exp \left[\frac{-{\left(x-a_2\right)}^{a_3}}{a_4}\right]\\ \begin{array}{cc}& \end{array}+a_5\exp \left[\frac{-{\left(x-a_6\right)}^{a_7}}{a_8}\right]+a_9\end{array}$$

The sum squared error function that fminsearch attempts to minimize is

$$F={\displaystyle \sum {[\log (\text{empirical BER})-\log (\text{fitted BER})]}^2}$$

The fitted BER points are the values in the output fitber, and the sum is over the E_b/N₀ points given in the input empEbNo. To avoid high-BER regions dominating the objective function, the sum squared equation uses the log of the BER values rather than the BER values themselves.

References

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

Version History

Introduced before R2006a