Using Fixed point maths in microcontrollers

Fixed point mathematics lets you calculate results that need to have a fractional multiplier and it lets you calculate results that would normally need floating point variables.  This saves a huge amount of memory.

For example if you want to calculate an ADC result that uses a 19.6mV step size then you need to multiply the ADC result by 0.0196.  Normal maths using a floating point variable would result in the following:

ADC reading Floating point calculation Floating point result
0 0 * 0.0196 0.0
127 127 * 0.0196 2.489
255 255 * 0.0196 4.998

With Fixed point mathematics you can do this quickly and efficiently using only integer type variables.

Note: Floating point variables use up huge amounts of resources in a microcontroller as microcontrollers are only really good at integer type variables.  To do floating point complex library functions are called up. This is similar to the old 386 processor (it too was no good at floating point operations being far too slow) so a separate floating point processor was used.

An example problem

Here is the problem to solve:

Using the minimum memory resources use an 8 bit ADC with 5V reference to transmit using a PIC micro serial data to a PC via the serial port the ADC reading every second.

The most important decision is not to use floating point - this saves lots of memory resource - all the rest; sending data to the serial port and timing will not take up much memory (the soft serial port (TX) that you can find here takes about 90 memory words).

So how to do it : First of all the step size of each ADC bit is:

5V/255 = 0.0196 (19.6mV)

Next work out the type of the variable that is big enough to hold the maximum expected value but is also the minimum size variable you can get away with. Factors such as the number of ADC bits and required calculation accuracy affect the variable size.

To convert, using fixed point, multiply the ADC reading by an integer scale factor but make sure the maximum value fits within the variable e.g If you choose a scale factor of 196 check the maximum value:

Maximum value : 255 * 196 = 49980

Here you can use a 16 bit variable (or unsigned int) as the maximum value is smaller than the maximum value that the 16 bit variable can hold.

Max value that unsigned int can store is 216-1 = 65535.

Since 49980 < 65535 the scheme is ok.

This has calculated the result using fixed point shown in the table below:

ADC reading Fixed point calculation Fixed point result
0 0 * 196 0.0
127 127 * 196 24892
255 255 * 196 49980

The decimal point is fixed four to the left - i.e. fixed point.

This has use the internal compiler routine '16 bit multiply' which is much simpler than a floating point multiply so you save loads of memory resource.

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