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Binary scaling
Binary scaling is a technique used mainly by embedded C, DSP and assembler programmers to perform a psuedo floating point using integer arithmetic. It is faster than floating point and is more accurate, but care must be taken not allow overflow.
A position for the virtual binary point is taken, and then subsequent arithetic operations determine the resultants binary point.
This obeys the mathematical laws of Exponentiation.
To give an example, a common way to use integer maths to simulate floating point is to multiply the co-effiecents by 65536. This is currently used in the microwindows utility nxcal to linearise the output of touchscreens.
This will place the binary point at B16.
For instance to represent 1.2 and 5.6 floating point real numbers as B16 one multiplies them by 2 ^ 16 giving
78643 and 367001
Multiplying these together gives
28862059643
To convert it back to B16 we need to divide it by 2 ^ 16.
This gives 440400B16, which when converted back to a floating point number (by diving again by 2 ^ 16, but holding the result as floating point) gives 6.71999. The correct floating point result is of course 6.72.
The scaling range here is for any number between 65535.9999 and -65536.0 with 16 bits to hold fractional quantities (of course assuming you have a 64 bit result register).Note that some computer architectures amy restrict arithmetic to 32 bit results. In this case extreme care must be taken not to overflow the 32 bit register. For other number ranges the binary scale can be adjusted for optimum accuracy.
Re-Scaling after Multiplication
The example above for a B16 multiplication is a simplified example. Re-scaling depends on both the B scale value and the word size. B16 is often used in 32 bit systems because it works simply by multipling and dividing by 65536 (or shifting 16 bits).
Consider the Binary Point in a 32 bit word thus:
0 1 2 3 4 5 6 7 8 9 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X
Placing the binary point at
- 0 gives a range of -1.0 to 0.999999.
- 1 gives a range of -2.0 to 1.999999
- 2 gives a range of -4.0 to 3.999999 and so on.
When using different B scalings the complete B scaling formula must be used.
Consider a 32 bit word size, and two variables, one with a B scaling of 2 and the other with a scaling of 4.
1.4 @ B2 is 1.4 * (2 ^ (wordsize-2-1)) == 1.4 * 2 ^ 29 == 0x2CCCCCCD
Note that here the 1.4 values is very well represented with 20 fraction bits ! This is why B scaling is always more accurate than floating point of the same wordsize.
Now a larger number 15.2 at B4.
15.2 @ B4 is 15.2 * (2 ^ (wordsize-4-1)) == 15.2 * 2 ^ 27 == 0x79999999A
Multiplying these 32 bit numbers give the 64 bit result 0x1547AE14A51EB852
This result is in B7 in a 64 bit word. Shifting it down by 32 bits gives the result in B7 in 32 bits.
0x1547AE14
To convert back to floating point, we need to divide this by (2^(wordsize-7-1)) == 21.2800000099
Various scalings maybe used. B0 for instance can be used to represent any number between -1 and 0.999999999.
Binary Angles
Binary angles are mapped using B0, with 0 as 0 degrees, 0.5 as 90 (or pi/4), -1.0 or 0.9999999 as 180 (or pi/2) and 270 as -0.5 (or 3.pi/2). When these binary angles are added using normal twos complement mathematics the rotation of the angles is correct, even when crossing the sign boundary (this of course does away with check like ((if >= 360.0) when handling normal degrees).
Application of Binary Scaling Techniques
Binary scaling techniques were used in the 1970's and 80's for real time computing that was mathematically intensive, such as flight simulation. The code was often commented with the binary scalings of the intermediate results of equations.
Binary scalling is still used in many DSP applications and custom made microprocessors are usually based on binary scaling techniques.
Binary scaling is currently used in the linux microwindows release to linearise touchscreens. It is also used in the DCT used to compress JPEG images in utilities such as the GIMP.
Although floating point has taken over to a large degree, where speed and extra accuracy are required, binary scaling is faster and more accurate.