Calculate better derivative of an signal?

Hello

I=92m sampling a signal at 1000Hz rate. I need to find the derivative of
the signal every 100ms.

I have tried to calculate it like this every 100ms:

dx =3D (x_old =96 x) / 0.1
x_old =3D x

x is the signal at that moment.

But I=92m thinking of how I could use the other 99 samples to get a
better approximation of the derivative over 100ms.

If I could do a least square fit of a straight line of 100 samples and
take the slope of that line as the derivative.

Any ideas?

On Apr 20, 4:29=A0pm, Lox <skolpojke...@yahoo.se> wrote:

>
> If I could do a least square fit of a straight line of 100 samples and
> take the slope of that line as the derivative.


You could exactly fit a polynomial of degree 99
to the 100 samples and then find the derivative
of that instead.  Is there a reason that you are
not telling us why least-squares fitting of a straight
line to 100 samples is a more appropriate solution?

--Dilip Sarwate

On 04/20/2011 02:29 PM, Lox wrote:
> Hello
>
> I�m sampling a signal at 1000Hz rate. I need to find the derivative of
> the signal every 100ms.
>
> I have tried to calculate it like this every 100ms:
>
> dx = (x_old � x) / 0.1
> x_old = x
>
> x is the signal at that moment.
>
> But I�m thinking of how I could use the other 99 samples to get a
> better approximation of the derivative over 100ms.
>
> If I could do a least square fit of a straight line of 100 samples and
> take the slope of that line as the derivative.
>
> Any ideas?

Define what "better" means within the context of the larger problem that 
you're trying to solve.  If the solution doesn't drop out, you'll at 
least be a lot closer to it.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

On Apr 20, 5:29=A0pm, Lox <skolpojke...@yahoo.se> wrote:
> Hello
>
> I=92m sampling a signal at 1000Hz rate. I need to find the derivative of
> the signal every 100ms.
>
> I have tried to calculate it like this every 100ms:
>
> dx =3D (x_old =96 x) / 0.1
> x_old =3D x
>
> x is the signal at that moment.
>
> But I=92m thinking of how I could use the other 99 samples to get a
> better approximation of the derivative over 100ms.
>
> If I could do a least square fit of a straight line of 100 samples and
> take the slope of that line as the derivative.
>
> Any ideas?

As Dilip sugested, try fitting a straight line to the data and finding
its slope. If the data is uniformly sampled, then the linear
regression becomes quite simple. Take a look at a paper I wrote a
while back.

http://www.claysturner.com/dsp/FIR_Regression.pdf

Regards,
Clay

On Apr 20, 11:29=A0pm, Lox <skolpojke...@yahoo.se> wrote:
> Hello
>
> I=92m sampling a signal at 1000Hz rate. I need to find the derivative of
> the signal every 100ms.
>
> I have tried to calculate it like this every 100ms:
>
> dx =3D (x_old =96 x) / 0.1
> x_old =3D x
>
> x is the signal at that moment.
>
> But I=92m thinking of how I could use the other 99 samples to get a
> better approximation of the derivative over 100ms.
>
> If I could do a least square fit of a straight line of 100 samples and
> take the slope of that line as the derivative.
>
> Any ideas?

Look up 'numerical differentiation'.

This task is a standard task in numerical maths, and your choise
of solution might get you into any sort of trouble, rangig from
the hoplessly inaccurate result, to the ridiculously computationally
expensive.

Rune

Hi Clay,

>On Apr 20, 5:29=A0pm, Lox <skolpojke...@yahoo.se> wrote:
>> Hello
>>
>> I=92m sampling a signal at 1000Hz rate. I need to find the derivative
of
>> the signal every 100ms.
>>
>> I have tried to calculate it like this every 100ms:
>>
>> dx =3D (x_old =96 x) / 0.1
>> x_old =3D x
>>
>> x is the signal at that moment.
>>
>> But I=92m thinking of how I could use the other 99 samples to get a
>> better approximation of the derivative over 100ms.
>>
>> If I could do a least square fit of a straight line of 100 samples and
>> take the slope of that line as the derivative.
>>
>> Any ideas?
>
>As Dilip sugested, try fitting a straight line to the data and finding
>its slope. If the data is uniformly sampled, then the linear
>regression becomes quite simple. Take a look at a paper I wrote a
>while back.
>
>http://www.claysturner.com/dsp/FIR_Regression.pdf
>
>Regards,
>Clay

That's a nice, clearly reasoned, paper. I've done similar things, but I
looked at what I was doing purely as a correlation with the pattern of
interest (a straight line in this case). I'd never followed through to see
that it works out exactly the same as a regression analysis.

Steve

On 21 Apr, 07:27, Clay <c...@claysturner.com> wrote:
> On Apr 20, 5:29=A0pm, Lox <skolpojke...@yahoo.se> wrote:
>
>
>
>
>
>
>
>
>
> > Hello
>
> > I=92m sampling a signal at 1000Hz rate. I need to find the derivative o=
f
> > the signal every 100ms.
>
> > I have tried to calculate it like this every 100ms:
>
> > dx =3D (x_old =96 x) / 0.1
> > x_old =3D x
>
> > x is the signal at that moment.
>
> > But I=92m thinking of how I could use the other 99 samples to get a
> > better approximation of the derivative over 100ms.
>
> > If I could do a least square fit of a straight line of 100 samples and
> > take the slope of that line as the derivative.
>
> > Any ideas?
>
> As Dilip sugested, try fitting a straight line to the data and finding
> its slope. If the data is uniformly sampled, then the linear
> regression becomes quite simple. Take a look at a paper I wrote a
> while back.
>
> http://www.claysturner.com/dsp/FIR_Regression.pdf
>
> Regards,
> Clay

Thank you. It worked really well :-)

On Apr 21, 12:27=A0am, Clay <c...@claysturner.com> wrote:

>
> As Dilip sugested, try fitting a straight line to the data and finding
> its slope.

I DID NOT suggest fitting a straight line to
the samples; that was the OP's idea.  He
began by wanting to estimate the derivative
of a signal at 100 ms intervals.  He had samples
of the signal taken at millisecond intervals.  He
then seemed to change his mind as to what he
wanted and said that he wanted the derivative
over the entire 100 ms interval.  What does this
mean?  100 values of the derivative at 1 millisecond
intervals? or the average value of the 100 derivatives
at 1 millisecond intervals? or simply using 100
samples to evaluate the derivative at one point?
No useful answer can be given to the "question"
being asked unless the question is stated more
clearly and the OP follows Tim Wescott's advice
to "Define what `better' means within the context
of the larger problem that you're trying to solve."

--Dilip Sarwate

Lox schrieb:
> Hello
>
> I�m sampling a signal at 1000Hz rate. I need to find the derivative of
> the signal every 100ms.
>
> I have tried to calculate it like this every 100ms:
>
> dx = (x_old � x) / 0.1
> x_old = x
>
> x is the signal at that moment.
>
> But I�m thinking of how I could use the other 99 samples to get a
> better approximation of the derivative over 100ms.
>
> If I could do a least square fit of a straight line of 100 samples and
> take the slope of that line as the derivative.
>
> Any ideas?

One idea would be to design a high pass filter which approximates the 
derivative (for example G(f)=abs(f)) and pass the signal through it.
But you have to check the math before if this is appropriate for your task.

Greetz,
Sebastian

On 4/20/2011 2:29 PM, Lox wrote:
> Hello
>
> I�m sampling a signal at 1000Hz rate. I need to find the derivative of
> the signal every 100ms.
>
> I have tried to calculate it like this every 100ms:
>
> dx = (x_old � x) / 0.1
> x_old = x
>
> x is the signal at that moment.
>
> But I�m thinking of how I could use the other 99 samples to get a
> better approximation of the derivative over 100ms.
>
> If I could do a least square fit of a straight line of 100 samples and
> take the slope of that line as the derivative.
>
> Any ideas?

Nobody so far mentioned that "differentiation is a noisy process" .. a 
rule of thumb at the very least.  Something to be aware of in system 
design involving differentiators.

It appears that Clay's approach might help avoid some of this but I got 
hung up on "differentiation" vs. "slope determination" and don't have 
the guts or time today to wade through the math.
I thought the comments about P-M differentiators in contrast were *very* 
interesting!

How about it Clay?  Some qualitative comments perhaps?  er .. re "noisy"?

Fred