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Hello everyone,

I was puzzled by the implementation of digital integrator for a while. Here 
are my quesions:

What is the meaning, or defination, of the time constant in the case of a 
digital integrator? For the analog one, it was defined by the RC network. 
For a 200ksps ADC input data, if the integration time is 1ms, I don't know 
which of the following is correct:

A) accumulate 200 points (N = 1ms/(1/200k)) of data, output one point, then 
process the next 200 input data, output another one. The data rate will 
change from the input 200k to output 1k.

B) accumulate continuously all the data, the result is divided by 200. It 
was like the direct translation of the integration formula, the data rate is 
not changed. In this case, the amplitude will be divided by the N but the 
time constant has nothing to do with the  filtering effect.

Another problem is the overflow of the accumulator. I'm thinking to caculate 
the average of the first 100 points of data, and substact it from all the 
subsequent input data to prevent the possible overflow. Any better ideas?

Thanks for reading,
JJ

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Steve wrote:
> Hello everyone,
> 
> I was puzzled by the implementation of digital integrator for a while. Here 
> are my quesions:
> 
> What is the meaning, or defination, of the time constant in the case of a 
> digital integrator? For the analog one, it was defined by the RC network. 
> For a 200ksps ADC input data, if the integration time is 1ms, I don't know 
> which of the following is correct:
> 
> A) accumulate 200 points (N = 1ms/(1/200k)) of data, output one point, then 
> process the next 200 input data, output another one. The data rate will 
> change from the input 200k to output 1k.
> 
> B) accumulate continuously all the data, the result is divided by 200. It 
> was like the direct translation of the integration formula, the data rate is 
> not changed. In this case, the amplitude will be divided by the N but the 
> time constant has nothing to do with the  filtering effect.
> 
Neither

Just a bare integrator doesn't have a time constant, whether it's 
digital or analog.

A bare integrator can have a _gain_, which in the case of an analog 
integrator has units of frequency, and in the simplest implementation 
with op-amps is, indeed, 1/RC.

One common style of expressing the gains in a PID loop is to use reset 
time = 1/(integrator gain), in which case the "reset time" is the "time 
constant" of the resulting zero.

If you absolutely must think of a "time constant" in terms of a digital 
integrating system, figure out what the analogous integrator gain is 
from the ADC to the DAC, and interpret that however you would interpret 
your analog system "time constant".

> Another problem is the overflow of the accumulator. I'm thinking to caculate 
> the average of the first 100 points of data, and substact it from all the 
> subsequent input data to prevent the possible overflow. Any better ideas?
> 
Eh?  Where are you using this integrator?  Arrange your system so it 
doesn't overflow, and be happy.  If it's a control system, chances are 
you need to implement some sort of integrator anti-windup.

What you are proposing won't prevent overflow in the end, but it _will_ 
make your system do odd things.

-- 

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

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

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Steve wrote:
> Hello everyone,
>
> I was puzzled by the implementation of digital integrator for a while. Here
> are my quesions:
>
> What is the meaning, or defination, of the time constant in the case of a
> digital integrator? For the analog one, it was defined by the RC network.
> For a 200ksps ADC input data, if the integration time is 1ms, I don't know
> which of the following is correct:
>
> A) accumulate 200 points (N = 1ms/(1/200k)) of data, output one point, then
> process the next 200 input data, output another one. The data rate will
> change from the input 200k to output 1k.
>
> B) accumulate continuously all the data, the result is divided by 200. It
> was like the direct translation of the integration formula, the data rate is
> not changed. In this case, the amplitude will be divided by the N but the
> time constant has nothing to do with the  filtering effect.
>
> Another problem is the overflow of the accumulator. I'm thinking to caculate
> the average of the first 100 points of data, and substact it from all the
> subsequent input data to prevent the possible overflow. Any better ideas?
>
> Thanks for reading,
> JJ

You mean a time constant 1/(1+sT). (not a pure integrator k/s). This
has a z-transform found from the Bilinear transform or otherwise.
There is no equivalent T in digital though it behaves the same way.

F.

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Steve wrote:

> What is the meaning, or defination, of the time constant in the case of a 
> digital integrator?

There is no such thing as time constant of an integrator.
The analog or digital integrator does have gain. In the analog case, the 
gain is 1/RC. In the digital case, the gain is Fs.


> For the analog one, it was defined by the RC network. 
> For a 200ksps ADC input data, if the integration time is 1ms, I don't know 
> which of the following is correct:
> 
> A) accumulate 200 points (N = 1ms/(1/200k)) of data, output one point, then 
> process the next 200 input data, output another one. The data rate will 
> change from the input 200k to output 1k.
> 
> B) accumulate continuously all the data, the result is divided by 200. It 
> was like the direct translation of the integration formula, the data rate is 
> not changed. In this case, the amplitude will be divided by the N but the 
> time constant has nothing to do with the  filtering effect.
> 
> Another problem is the overflow of the accumulator. I'm thinking to caculate 
> the average of the first 100 points of data, and substact it from all the 
> subsequent input data to prevent the possible overflow. Any better ideas?

Neither of the above mentioned does make any sense. It all depends on 
what exactly are you trying to accomplish, however you seem to miss the 
understanding of the basics. Get a textbook on digital signal processing.


Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

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I made a real time digital integrator with a 1st order IIR filter:
Y[n]= X[n]+ Ki * Y[n-1]
Ki is the coeficient of a low pass filter. The filter corner frequency
must be 10 times lower than the minimum frequency that you need to
integrate.
This is not an ideal integrator, an analog RC integrator either.
It is possible that you need a high pass filter before and after to
integrate.
I don't know for what you want to use it.
Ask me for more information.

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Steve,

I would think that a RC low pass filter can be modeled as a first
order Butterworth filter with the -3dB cutoff frequency fc = 1/
(2*pi*R*C).

Also, some details on recursive filter implementation is available
@ http://www.dspguide.com/CH19.PDF

HTH,
Krishna
http://dsplog.blogspot.com

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