Re: integration of a continuous function



Alex_001 wrote:

Here's my problem:

we're aquiring a signal form a piezoelectric force sensor; the analog
signal is filtered by a lowpass filter at 5 kHz and sampled at 100 kHz. In
such a way we get a "smooth" waveform, that is what we want.

Keep in mind the sensor has a frequency response that approximates force in a
certain range of frequencies. For instance it may be from 20 to 20KHz the
response is more or less linear with the square of the frequency. How the sensor
is mounted affects this frequency response. So you can only accurately determine
the response in situ.


What we need is to calculate the integral of this signal, on a 5 ms
window.

What you should consider is using a filter that is inverse of the frequency
response of your system. That will look a lot like integration.


I was wondering if there's a EASY way to reduce the sampling frequency and
still get an accurate estimate (there are surely ways to do that, but if
too compicated is useless for our application, we'd rather have a higher
sampling frequency).

Is higher frequency above the point where the sensors response is no longer
linear with the frequency squared?

-jim



Following illywhacker's (and someoneelse's) advice, I wrote my continuous
function as a linear combination of shifted sinc functions weighted by the
sample values (recostruction theorem).
Denoting with X(t) the continuous function, and X(n*T) it samples in [0,
To], I got for the integral in [0 T0] (N = total number of samples):

integral[0 , T0]{X(t)dt} = sum[n=0..N]{X(nT)* integral[-n ,
(T0-nT)/T]{sinc(p)dp}}

Now, the function integral[0 , x]{sinc(x)dx} is known and tabulate (Matlab
implements such a function, for example). So, if my procedure was correct,
it should be not that hard...I will try and let you know.

Alex
.

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