Binary scaling: Difference between revisions

Revision as of 19:49, 5 January 2007

Latest revision as of 22:52, 22 December 2022

(127 intermediate revisions by 79 users not shown)

Line 1:

{{Mergeto|#REDIRECT [[Fixed-point arithmetic|date=December#Binary scaling]] {{R from merge}} {{R to section}} {{R with 2006possibilities}}

{{Wikify|October 2006}}

'''Binary scaling''' is a [[computer programming]] technique used mainly by embedded [[C programming language|C]], [[Digital signal processing|DSP]] and [[assembly language|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'.

Binary points obey 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 [[touchscreen]]s.

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 may 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 30 fraction bits ! A 32 bit [[IEEE floating-point standard|real number]] has 23 bits to store the fraction in. This is why B scaling

is always more accurate than floating point of the same wordsize.

This is especially useful in [[integrator]]s or repeated summing of small quantities where [[rounding error]]

can be a subtle but very dangerous problem, when using [[floating point]].

Now a larger number 15.2 at B4.

15.2 @ B4 is 15.2 * (2 ^ (wordsize-4-1)) == 15.2 * 2 ^ 27 == 0x79999999A

Again the number of bits to store the fraction is 28 bits.

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).

[[Image:Binary angles.png|250px|thumb|Binary Angles Diagram]]

== 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 [[digital Signal Processing|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 [[touchscreen]]s.

It is also used in the [[Discrete cosine transform|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.

[[Category:Computing terminology]]

[[Category:Computer programming]]