Re: help! how do I formulate this sampling system and analyze it?
- From: Jerry Avins <jya@xxxxxxxx>
- Date: Mon, 30 Jan 2006 22:22:22 -0500
lucy wrote:
Hi all,
This system consists of an DAC, a processor, and a ADC. The flow graph is like as follows:
input-digital-signal -> DAC -> filter(Continuous Time) -> ADC -> output-digital-sigal
That's a weird way to do things. Is it an academic exercise?
The problem is that the DAC is a zero order hold, and the ADC is more strange:
the ADC samples the continuous time waveform by integrate the input signal in a period T, then put that integral result as the sample at that time index.
Most DACs have zero-order holds. A DAC must be followed by a reconstruction filter, which smooths out the steps. Your analog filter can fulfill that function while it does its other stuff.
Many low-speed high precision ADCs used to be integrating types. It is usually sufficient to think of the sampling instant as the middle of the integration interval. The need to remove from the ADC input components of the signal more than half the sampling frequency makes that approximation quite accurate. The integration rejects noise frequencies that are a multiple of the sample rate. It was common practice to run these converters at the power-line frequency. That made them insensitive to line noise and its harmonics. When implemented with counters and comparators, integrating ADCs have inherent linearity. Dual and quad slope converters null out other sources of inaccuracy.
As an example, the ADC integrates the input signal from (n-0.5)T to (n+0.5)T, and then use the integration result as the output sample y[n], which is in discrete time.
Now I need to formulate this system in DSP, and analyze the result, to see how much does it deviate from a second system(given the same input, how much does the output deviate, can they be made to be exactly the same output???), which is as follows:
input-digital-signal -> filter(Discrete Time) -> output-digital-sigal
I don't know what filters you can use that would be identical even without complications. If you oversample sufficiently, you can come very close.
----------------------------
My headache is that the presence of non-traditional ADC/DAC complicated the matters...
Your DAC is typical. How else would you have it behave? As for the ADC, remember: there are no Dirac impulses in real life. The output of every ADC or sample-and-hold represents an integration over the aperture time. Integrating converters simply extend the integration time to the full sample period. Don't sweat it. At the oversampling rate needed to match the digital and analog filters closely in overall response, the signal curvature during the sample time will be very small.
Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ .
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