Desired Sum of Two Curves; suggestions?

From: axlq (axlq_at_spamcop.net)
Date: 10/19/04 

Date: Tue, 19 Oct 2004 04:07:12 +0000 (UTC)

I'm working on an adaptive digital filter that requires two
nonlinear control parameters, call them b and c. The filter itself
controls these parameters. I'm trying to invent formulas for b
and c (as functions of another parameter x) that conform to the
following conditions:

0. I don't care about negative x values.

1. The sum b(x)+c(x) results in a curve that rises from zero at
x=0, always has a positive slope with no inflection points, and
asymptotically approaches a value g -- Sort of similar to the curve
g*(1-exp(-x)) but doesn't have to be that specific curve.

2. b(x) must rise from zero at x=0 to some maximum, then fall
through one inflection point toward zero again but never reach it,
asymptotically approaching it. Say the maximum occurs at x=p, and
c(x)=0 at x=infinity.

3. c(x) must start out at zero at x=0 and sort of crawl along the x
axis until x>p, after which c(x) rises asymptotically toward value
g. That is, c(x) rises from zero, always has a positive slope, but
has one inflection point and reaches a maximum value at x=infinity.

I just need some generic functions I can play with; the details
will have to be worked out in the context of the data stream I'm
filtering.

<pre>
If you have a fixed width font, then graphically, a(x) looks like
this:

 | *
 | * *
 | * *
 | * *
 | * *
 |* * *
 *--------p------------------------->x

b(x) would look like this:

 | * *
 | *
 | *
 | *
 | *
 | *
 *--------p-------------------------->x

And the sum a(x)+b(x) would look like this:

 | * *
 | *
 | *
 | *
 | *
 |*
 *----------------------------------->x
</pre>

Does anyone know of a family of curves that will do this?

I can come up with something for a(x). Start with a parabola that
intersects the x axis at zero and 2p. Then make the parabola a
function of 2p*(1-exp(-x)) instead of a function of x. That way the
"independent variable" of the parabola never reaches 2p, and you get
a curve that rises and falls, but never crosses the x axis a second
time.

I can't come up with a function for an s-shaped curve such as b(x)
though, which rises from zero, always has a positive slope with one
inflection point, and asymptotically approaching some maximum value.

Any ideas?

-Alex
P.S. Before someone makes a wrong assumption, this is not homework.
I got my master's degree over a decade ago. This is related to my
job.

Relevant Pages

  • Re: Desired Sum of Two Curves; suggestions?
    ... > I'm working on an adaptive digital filter that requires two ... The sum b+cresults in a curve that rises from zero at ... squared error of Ea and the squared error of Eb. ...
    (sci.math)
  • Re: Desired Sum of Two Curves; suggestions?
    ... The sum b+cresults in a curve that rises from zero at ... > through one inflection point toward zero again but never reach it, ...
    (sci.math)
  • Re: Curving An Object Ball(Part VI-Letting it All Hang Out))
    ... and none are equal to zero. ... Bob was NOT agreeing that there is any observable curve, ... very short distance of the cue ball. ... Even though bob is wrong about my method because my shot has nothing ...
    (rec.sport.billiard)
  • Re: Bobs shot 3 video
    ... I still think deviation from a straight line is zero, ... an 8 foot shot, at slow speed. ... As for whether its the draw, follow or sidespin that makes it ... curve, I still cant say for sure. ...
    (rec.sport.billiard)
  • Re: Zeroing load cell data
    ... time record-- zero load, followed by placing a known load on and leaving it ... -different- zero. ... You referred to a force/voltage curve. ... The two calibration points would be at one of the zeros, ...
    (sci.engr.control)