On Sat, 23 Apr 2005, Mike Ryan wrote:

> Just wondering if you could help me out,
>
> I need to perform basic integration and basic differentiation using 8 
> bit algorithms in assembler,
>
> Would you know the basic approach or where I could find out how to do it,

The differential of a vector of samples taken at constant intervals is a 
vector that contains the difference between adjacent samples in the 
input vector, for each pair of samples. The first output sample will be 
missing (it's a point of discontinuity so the differential is 
undefined). So:

diff[0] = NaN;
for i=1; i<N; ++i
 	diff[i] = input[i] - input[i-1];

Each diff[i] is the slope of the line defined by the points input[i] and 
input[i-1], with ordinate length 1. The differential vector is offset 
from the original vector by 1/2 sample to the right on the abscissa (X 
or t axis).

The integral of a vector of samples as above can be described otherwise 
like: the input vector is the differential vector of the integral 
vector of the input. So, from above:

 	input[i] = int[i] - int[i-1];

The missing part is 'c' the undefined part of the integral, and we need 
to supply it before starting. So int[0] = input[i] + c, int[i] = 
input[i] + int[i-1], ... . Note that we add a lot and the potential for 
overflow is huge (and you really want to check for it and limit the 
integral function or something similar):

c = 2^7; // for example
int[0] = input[0] + c;
for i=1; i<N; ++i
 	int[i] = input[i] + int[i-1];

Again as above the int[i] vector is offset wrt to the input abscissa by 
1/2 sample to the left.

The offset problem can be fixed for example by oversampling to 2 x the 
sample rate and interpolation (which can be simple or complex). It 
automatically increases the accuracy of the int[] and diff[] vectors, if 
the interpolation function is well choosen.

These forms be rewritten to operate on a stream if needed (remove the 
loop and execute a step with every input event or interrupt or stream 
byte or what you need). Note that if c=0 then int[N-1] is the area under 
the graph described by input[0..N] if you remove the sign from input[i] 
before adding it in.

Peter
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