Calculate better derivative of an signal?

On Apr 21, 8:40=A0am, dvsarwate <dvsarw...@yahoo.com> wrote:
> 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. =A0He
> began by wanting to estimate the derivative
> of a signal at 100 ms intervals. =A0He had samples
> of the signal taken at millisecond intervals. =A0He
> then seemed to change his mind as to what he
> wanted and said that he wanted the derivative
> over the entire 100 ms interval. =A0What does this
> mean? =A0100 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

True you did not suggest fitting the straight line. I should have
phrased my statement better - sorry about that. I find often when one
is asking questions, that he may have not completely framed his
problem yet so I allow for a little latitude on the answer since the
OP is still defining his problem. So our answers kind of shotgun
around the OP's yet to be determined answer. This kind of reminds me
of a friend when he needed to build something he would stare at his
large collection of junk because it would get him to thiking about
what he really needed.

Clay

On Apr 21, 5:34=A0am, "steveu" <steveu@n_o_s_p_a_m.coppice.org> wrote:
> Hi Clay,
>
>
>
>
>
>
>
> >On Apr 20, 5:29=3DA0pm, Lox <skolpojke...@yahoo.se> wrote:
> >> Hello
>
> >> I=3D92m sampling a signal at 1000Hz rate. I need to find the derivativ=
e
> of
> >> the signal every 100ms.
>
> >> I have tried to calculate it like this every 100ms:
>
> >> dx =3D3D (x_old =3D96 x) / 0.1
> >> x_old =3D3D x
>
> >> x is the signal at that moment.
>
> >> But I=3D92m 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 se=
e
> that it works out exactly the same as a regression analysis.
>
> Steve- Hide quoted text -
>
> - Show quoted text -

Steve,

Thanks, What put me on the path to this method was a problem where I
was demodulating iDEN which uses four 16-QAM subcarriers FDMed
together in a nearly 20kHz wide channel where the pilot codes were
nonuniformly spaced. And with noise in my receiver, I was forced into
a statisical approach. I later used the polarity method to achieve
frame lock to the ghost cancellation reference signal in analog tv. It
proved to be very quick and reliable. I thought it was kind of neat to
combine correllation with regression. It leaves you feeling you are on
the right track.

Clay

On Apr 21, 11:50=A0am, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 4/20/2011 2:29 PM, Lox 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?
>
> Nobody so far mentioned that "differentiation is a noisy process" .. a
> rule of thumb at the very least. =A0Something 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? =A0Some qualitative comments perhaps? =A0er .. re "noi=
sy"?
>
> Fred- Hide quoted text -
>
> - Show quoted text -

Fred,

In terms of qualitative answers, I've often found that low order
differentiators often work better in practice since the high orders
tend to magnify noise even more than the low order ones. High order
processes often rely heavily on delicately balanced cancellations. And
some processes will magnify the errors from such. Of course this is a
rule of thumb and to be sure there will be exceptions. For example
Lagrangian interpolation (polynomial fitting) in the classic
straightforward implementation is fraught with numerical issues but
the barycentric form uses ratios of differences so the errors tend to
cancel.  One of the comments I put into the article has to do with the
coefs themselves. A high order polynomial or trigonometric
differentiator tends to cluster most of its "energy" or computational
work on the central coefs whereas the slope filter coefs tends to put
more of the computational work away from the center. In terms of
additive noise this seems to lower sensitivity because noise needs to
have greater relative (to the signal) strength to affect the slope. I
haven't spent much time doing any quantitative noise analysis on this
approach. Maybe one day I'll revisit it.

Clay

On 4/21/2011 9:40 AM, Clay wrote:
> On Apr 21, 11:50 am, Fred Marshall<fmarshallxremove_th...@acm.org>
> wrote:
>> On 4/20/2011 2:29 PM, Lox wrote:
>>
>>
>>
>>
>>
>>> Hello
>>
>>> I&#4294967295;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 &#4294967295; x) / 0.1
>>> x_old = x
>>
>>> x is the signal at that moment.
>>
>>> But I&#4294967295;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- Hide quoted text -
>>
>> - Show quoted text -
>
> Fred,
>
> In terms of qualitative answers, I've often found that low order
> differentiators often work better in practice since the high orders
> tend to magnify noise even more than the low order ones. High order
> processes often rely heavily on delicately balanced cancellations. And
> some processes will magnify the errors from such. Of course this is a
> rule of thumb and to be sure there will be exceptions. For example
> Lagrangian interpolation (polynomial fitting) in the classic
> straightforward implementation is fraught with numerical issues but
> the barycentric form uses ratios of differences so the errors tend to
> cancel.  One of the comments I put into the article has to do with the
> coefs themselves. A high order polynomial or trigonometric
> differentiator tends to cluster most of its "energy" or computational
> work on the central coefs whereas the slope filter coefs tends to put
> more of the computational work away from the center. In terms of
> additive noise this seems to lower sensitivity because noise needs to
> have greater relative (to the signal) strength to affect the slope. I
> haven't spent much time doing any quantitative noise analysis on this
> approach. Maybe one day I'll revisit it.
>
> Clay

Thanks Clay,  I'm sure this will be helpful to folks interested in this. 
  I had noticed the comment about the coefs.

Fred

On 04/20/2011 05:29 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?

The analytical, discrete-time derivative has impulse response

   h[n] = cos(n*pi) / n      [proakis]

This filter is non-causal and infinite. However, by truncating to a
finite number of terms and introducing a delay, a causal FIR version
can be derived.

@BOOK{proakis,
   title = "{Digital Signal Processing: Principles, Algorithms, and Applications}",
   author = "John~G.~Proakis and Dimitris~G.~Manolakis",
   publisher = "Prentice Hall",
   edition = "third",
   year = "1996"}

-- 
Randy Yates                      % "So now it's getting late,
Digital Signal Labs              %    and those who hesitate
mailto://yates@ieee.org          %    got no one..."
http://www.digitalsignallabs.com % 'Waterfall', *Face The Music*, ELO

On Apr 26, 4:05&#4294967295;am, Randy Yates <ya...@ieee.org> wrote:
> On 04/20/2011 05:29 PM, Lox wrote:
>
>
>
>
>
> > Hello
>
> > I&#4294967295;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 &#4294967295; x) / 0.1
> > x_old = x
>
> > x is the signal at that moment.
>
> > But I&#4294967295;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?
>
> The analytical, discrete-time derivative has impulse response
>
> &#4294967295; &#4294967295;h[n] = cos(n*pi) / n &#4294967295; &#4294967295; &#4294967295;[proakis]
>
> This filter is non-causal and infinite. However, by truncating to a
> finite number of terms and introducing a delay, a causal FIR version
> can be derived.

Picking nits: The statement "*a* causal FIR version" might
be interpreted to indicate that the problem has exactly one
solution. It does not.

As I indicated a few days ago, the ideal differentiator can be
approxmated in any number of ways, depending on constraints
and requirements.

Rune

Randy Yates wrote:


> The analytical, discrete-time derivative has impulse response
> 
>   h[n] = cos(n*pi) / n      [proakis]

But this is simply wrong.

How about the signal x(n) = n ?


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

On 4/25/2011 7:44 PM, Vladimir Vassilevsky wrote:
>
>
> Randy Yates wrote:
>
>
>> The analytical, discrete-time derivative has impulse response
>>
>> h[n] = cos(n*pi) / n [proakis]
>
> But this is simply wrong.
>
> How about the signal x(n) = n ?
>
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Design Consultant
> http://www.abvolt.com

How about "the analytical, discrete-time differencer has impulse 
response...... " ?
'cause a "derivative" isn't a system but a differentiator or a 
differencer is.  And, the signal and the system are independent so apply 
x(n)=n to it .. fine.  No?

Fred

Fred Marshall wrote:
> On 4/25/2011 7:44 PM, Vladimir Vassilevsky wrote:
> 
>>
>>
>> Randy Yates wrote:
>>
>>
>>> The analytical, discrete-time derivative has impulse response
>>>
>>> h[n] = cos(n*pi) / n [proakis]
>>
>>
>> But this is simply wrong.
>>
>> How about the signal x(n) = n ?
>>
> 
> How about "the analytical, discrete-time differencer has impulse 
> response...... " ?
> 'cause a "derivative" isn't a system but a differentiator or a 
> differencer is.  And, the signal and the system are independent so apply 
> x(n)=n to it .. fine.  No?

No.

VLV

On 04/25/2011 10:44 PM, Vladimir Vassilevsky wrote:
>
>
> Randy Yates wrote:
>
>
>> The analytical, discrete-time derivative has impulse response
>>
>> h[n] = cos(n*pi) / n [proakis]
>
> But this is simply wrong.
>
> How about the signal x(n) = n ?

Ha! Good point (again) Vlad! I tried analyzing this
and got a series that doesn't converge.
-- 
Randy Yates                      % "So now it's getting late,
Digital Signal Labs              %    and those who hesitate
mailto://yates@ieee.org          %    got no one..."
http://www.digitalsignallabs.com % 'Waterfall', *Face The Music*, ELO