Re: help! how do I formulate this sampling system and analyze it?

lucy wrote:

Hi Tim,

This is really a great method! All of a sudden I've got enlightened a
lot, I feel. Thank you so much!

Tim Wescott wrote:

If you split the ADC up into an integrating part and a sampling part,
then the integrating part is a continuous-time function who's transfer
function is



Here is my headache part, I keep having difficulty recasting this
integral type ADC into an integration part(which can be deemed as a
convolutional filter), and a sampling part...

In this way, I can handle this ADC in traditional way that is most
commonly and thoroughly taught in DSP classes.

But what should be the integration convolution kernel( the filter)?

A boxcar that is 1/T for all |t| < T/2 and zero otherwise. 

The ADC integrate the input signal against which kernel then sample?



Assuming that you define y_n as appearing at t = nT this is a non-causal
system.  I don't believe that, so I'm going to assume that y_1 appears
at time T/2, y_2 appears at 3T/2, etc.  You can modify your math to be
non-causal if you feel like it, but remember to be very careful with
your notation.



You thought deeper than me.

All I know is that in the Matlab discrete time representation, the
system's prosessor/filter is centered at zero, but in Matlab
representation, I have moved it to be centered at (N+1)/2, where N is
the length of the filter after cutoff(the filter itself is of infinite
support) and N is an odd number.

So in fact, in Matlab direct implementation, the system is as simple
as:

digitial_spatial_waveform_signal_input  => conv(input, filter), where
the filter is as described above ->
digital_filtered_spatial_waveform_signal_output

Does this work well with your method?

It appears you're deriving an FIR filter. My method, being exact, will give you an IIR filter for an IIR filter.

Ta da! 


This is really cool.

In fact, I am guessing that if I keep getting deviation of the paper's
results from Matlab discrete-time implementation's result. I can
construct an "equivalent" discrete time filter in Matlab to put in:

digitial_spatial_waveform_signal_input  => conv(input, filter), where
the filter is as described above ->
digital_filtered_spatial_waveform_signal_output

and try to make their result match.

Now let's think reversely, given a discrete-time filter which is used
in "conv" in Matlab,

how to use your above method to devise a continuous time
processor/filter to put tinto the miexed signal system:

digitial_spatial_waveform_signal_input  => DAC (zero order hold type
with no reconstruction filter) => continuous time processor/filter, but
still implemented in Matlab -> ADC(integral and dump type) =>
digital_filtered_spatial_waveform_signal_output

Theoratically, will these two discrete-time system and mixed signal
system produce exactly same result? If so, how to obtain the continuous
time processor from a given discrete time filter using your method?

Thanks a lot!



--

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

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