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fixpt_look1_func_approx - Optimize fixed-point approximation of nonlinear function by interpolating lookup table data points

Syntax

[xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,nptsmax) [xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[]) [xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,[],nptsmax) [xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydtydt,yscale,rndmeth,errmax,nptsmax,spacing)

Description

[xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,nptsmax) returns the optimal breakpoints of a lookup table, an ideal function applied to the breakpoints, and the worst-case approximation error. The lookup table satisfies the maximum acceptable error and maximum number of points that you specify.

[xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[]) returns the optimal breakpoints of a lookup table, an ideal function applied to the breakpoints, and the worst-case approximation error. The lookup table satisfies the maximum acceptable error that you specify.

[xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,[],nptsmax) returns the optimal breakpoints of a lookup table, an ideal function applied to the breakpoints, and the worst-case approximation error. The lookup table satisfies the maximum number of points that you specify.

[xdata,ydata,errworst] = fixpt_look1_func_approx('func',... xmin,xmax,xdt,xscale,ydtydt,yscale,rndmeth,errmax,nptsmax,spacing) returns the optimal breakpoints of a lookup table, an ideal function applied to the breakpoints, and the worst-case approximation error. The lookup table satisfies the maximum acceptable error, maximum number of points, and breakpoint spacing that you specify.

In each case, fixpt_look1_func_approx interpolates between lookup table data points to optimize the fixed-point approximation. The inputs xmin and xmax specify the range over which to approximate the breakpoints. The inputs xdt, xscale, ydt, yscale, and rndmeth follow conventions used by fixed-point Simulink blocks.

The inputs errmax, nptsmax, and spacing are optional. Of these inputs, you must specify at least errmax or nptsmax. If you omit one of those two inputs, you must use brackets, [], in place of the omitted input. fixpt_look1_func_approx ignores that requirement for the lookup table.

If you do not specify spacing, and more than one spacing satisfies errmax and nptsmax, fixpt_look1_func_approx chooses in this order: power-of-2 spacing, even spacing, uneven spacing. This behavior applies when you specify both errmax and nptsmax, but not when you specify just one of the two.

Inputs

func

Function of x for which to approximate breakpoints. Enclose this expression in single quotes, for example, 'sin(2*pi*x)'.

xmin

Minimum value of x.

xmax

Maximum value of x.

xdt

Data type of x.

xscale

Scaling for the x values.

ydt

Data type of y.

yscale

Scaling for the y values.

rndmeth

Rounding mode supported by fixed-point Simulink blocks:

`'Ceiling'`	Round to the nearest representable number in the direction of positive infinity.
`'Floor'` (default)	Round to the nearest representable number in the direction of negative infinity.
`'Nearest'`	Round to the nearest representable number.
`'Toward Zero'`	Round to the nearest representable number in the direction of zero.

errmax

Maximum acceptable error between the ideal function and the approximation given by the lookup table.

nptsmax

Maximum number of points for the lookup table.

spacing

Spacing of breakpoints for the lookup table:

'even' Even spacing

'pow2' Even, power-of-2 spacing

'unrestricted' (default) Uneven spacing

If you specify... The breakpoints of the lookup table...

If you specify...	The breakpoints of the lookup table...
`errmax` and `nptsmax`	Meet both criteria, if possible. The `errmax` requirement has higher priority than `nptsmax`. If the breakpoints cannot meet both criteria with the specified spacing, `nptsmax` does not apply.
`errmax` only	Meet the error criteria, and `fixpt_look1_func_approx` returns the fewest number of points.
`nptsmax` only	Meet the points criteria, and `fixpt_look1_func_approx` returns the smallest worst-case error.

errmax and nptsmax

Meet both criteria, if possible.

The errmax requirement has higher priority than nptsmax. If the breakpoints cannot meet both criteria with the specified spacing, nptsmax does not apply.

errmax only

Meet the error criteria, and fixpt_look1_func_approx returns the fewest number of points.

nptsmax only

Meet the points criteria, and fixpt_look1_func_approx returns the smallest worst-case error.

Outputs

`xdata`	Vector of breakpoints for the lookup table.
`ydata`	Vector of values from applying the ideal function to the breakpoints.
`errworst`	Worst-case error, which is the maximum absolute error between the ideal function and the approximation given by the lookup table.

Examples

Approximate a fixed-point sine function using a lookup table:

func = 'sin(2*pi*x)';
% Define the range over which to optimize breakpoints
xmin = 0;
xmax = 0.25;
% Define the data type and scaling for the inputs
xdt = ufix(16);
xscale = 2^-16;
% Define the data type and scaling for the outputs
ydt = sfix(16);
yscale = 2^-14;
% Specify the rounding method
rndmeth = 'Floor';
% Define the maximum acceptable error
errmax = 2^-10;
% Choose even, power-of-2 spacing for breakpoints
spacing = 'pow2';
% Create the lookup table
[xdata,ydata,errworst] = fixpt_look1_func_approx(func,...
 xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[],spacing);