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PIC Microcontoller Radix Math Method

5 digit BCD unpacked to 16bit binary in 33 inst, 33 cycles by Peter Hemsley

Published in EPE Dec '03 issue. By PETER HEMSLEY

Speed up your PIC's data conversion and compression.

MATHS is all about playing with numbers, so let's play and create a super-fast decimal to 16-bit binary routine for PIC microcontrollers. It is based on expressing powers of ten in terms of powers of two.

Here are the first five powers of ten: 

1     = 1
10    = 8 + 2
100   = 64 + 32 + 4
1000  = 512 + 256 + 128 + 64 + 32 + 8
10000 = 8192 + 1024 + 512 + 256 + 16
If X represents any decimal digit between 0 and 9 then: 
X     = (X * 1)
X0    = (X * 8) + (X * 2)
X00   = (X * 64) + (X * 32) + (X * 4)
X000  = (X * 512) + (X * 256) + (X * 128) + (X * 64) + (X * 32) + (X * 8)
X0000 = (X * 8192) + (X * 1024) + (X * 512) + (X * 256) + (X * 16)
These five expressions are the basis on which we can write a routine to convert a string of decimal digits into binary. The routine is to be written in assembler so any expression must be conducive to available processor instructions, namely NIBBLE SWAP, SHIFT LEFT and ADD.

Conducive numbers to work with are: 

2   (SHIFT LEFT to multiply by 2)
16  (NIBBLE SWAP to multiply by 16)
256 (The result goes into the high byte of the binary)
If your mind is somewhat blank at this point let's try a simple example to get the grey matter working.

The number 300 can be written as: 

(3 * 64) + (3 * 32) + (3 * 4)
Re-writing it in terms of conducive numbers we get: 
3 * 16 * 2 * 2 + 3 * 16 * 2 + 3 * 2 * 2
Now reduce and re-arrange the expression to: 
((3 * 16 + 3) * 2 + 3 * 16) * 2 = 300
Check it with your calculator.

This expression can be calculated easily using NIBBLE SWAP, SHIFT LEFT and ADD.

Ok that was easy enough, so now for the tricky part.

The 1000's and 10000's expressions contain six and five terms respectively. If we also use SUBTRACT, the number of terms can be reduced to three and four: 

X     = (X * 1)
X0    = (X * 8) + (X * 2)
X00   = (X * 64) + (X * 32) + (X * 4)
X000  = (X * 1024) - (X * 32) + (X * 8)
X0000 = (X * 8192) + (X * 2048) - (X * 256) + (X * 16)
These five expressions can now be written in terms of conducive numbers and combined to give: 
N = (((D1 + D3 + D4 * 256) * 2 + D2 * 16 + D2 + D3 * 256) * 2 - D3 * 16 +
D2 * 16 + D1 + D4 * 16 * 256) * 2 + D4 * 16 + D0 - D4 * 256
Where D0 = ones, D1 = tens, D2 = hundreds, D3 = thousands, D4 = ten thousands

To save a lot of typing, and you a big headache, the details of how this expression was arrived at have been omitted. It is simple enough though a little lengthy.

There is a problem however, it is the -D4 * 256 at the end of the expression. If the input is greater than 63231 the running total will exceed the allotted 16 bits. So, again, re-arrange the expression. 

N = (((D1 + D3 + D4 * 256) * 2 + D2 * 16 + D2 + D3 * 256) * 2 - D3 * 16 +
D2 * 16 + D1) * 2 + D4 * 16 + D0 - D4 * 256 + D4 * 16 * 256 * 2
Now there is an addition of a large number at the end of the expression, therefore overflow will not occur.

The PIC routine in Listing 1 is an almost literal translation of this expression into assembler, with just a few tweaks to make the code more efficient. The variables may be allocated (equated) to any registers of your choice.

Incidentally, the routine will work with numbers up to 99999, the 17th bit (or bit 16) being returned in the carry.

Finally, if your numerical value is expressed in ASCII characters, each character may be converted to a BCD (binary-coded-decimal) format by subtracting 48, which makes it easy to then check if it is a valid decimal digit.

[xh]RESOURCE This software routine is available from the [i]EPE PCB Service[r] on 3.5in disk (Disk 6, [i]PIC Tricks[r] folder), for which a nominal handling charge applies. It is also available for free download from the [i]EPE[r] Downloads page, accessible via the home page at [bo]www.epemag.wimborne.co.uk[r]. It is in the [i]PIC Tricks[r] folder, as file [bo]Dec2Bin16.txt[r].

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