Clearer Response to Polymath's comp.dsp - a FAQ Post
- From: "dbell" <dbell@xxxxxxxxxx>
- Date: 15 Oct 2005 16:40:17 -0700
Here is an attempt at a slightly more precise response to Polymath's
post. I am only clarifying my response to his first question/comment
since that is the part of my response that initiated so much negative
feedback.
I think (still) Polymath's comment about the location of 'T' has
some validity, AS I READ HIS COMMENTS, which is not holding him to
strict mathematical rigor but interpreting what he said in terms of his
'objection' (which is the way I read a lot of the postings here).
As I read it, his comment is about a 'preferred' (by him) modified
version of the sampling explanation that rbj presented. Polymath's
objection was that he felt that the scaling of the modulated impulse
train representing the input signal should not depend on the sample
rate, in the same way that an A/D converter output would not be
expected to. So if one had synchronously impulse sampled a signal with
two different samplers, one sampling every T0 seconds, and the other
every 2*T0 seconds, the scaled output impulses that corresponded to the
simultaneous sampling impulses would be scaled the same. I realize that
analog impulse sampling and A/D sampling are not equivalent, but they
have some similarity that permits comparison, which seems to be the
basis of Polymath's preference.
Hopefully the following comments about implementing Polymath's
preferred explanation will be totally clear.
Define 'M', related to 'T' (sample period in seconds) by the
relationship
T= M seconds
That would make 'M' a dimensionless scalar, which I use below.
Assuming that the explanation given by rbj is correct, which I believe,
it can be modified as follows:
1) Scale rbj's definition of the sampling function q(t) by (1/M).
This will result in a revised x(t)*q(t) that does not depend on the
value of 'T', as Polymath would prefer. This result has a some
similarity to what comes out of a real-world A/D.
2) Then, to maintain perfect reconstruction for a band limited signal,
as is done in rbj's explanation, scale the gain of the ideal analog
reconstruction filter by 'M'. This makes the filter gain depend on
'T' (through the application of 'M'), which rbj and apparently
many others do not prefer. In fairness, it is not consistent with a
real-world anti-imaging filter that would follow a zero-order hold D/A,
but it is consistent with explanations of digital interpolation filters
that are used on zero-padded data.
It is all equivalent. No units/dimensions changed ANYWHERE!
That's all, folks.
Dirk
DISCLAIMER: Having posted this I will not respond to any objections
about any related previous posts that I have made. My sincere apologies
for not being clear in what I said the first time (God, I hope I was
this time). My apologies to both jbr and rbj for the misuse of jbr's
initials. Finally, Polymath, if I have put any words in your mouth that
you do not agree with, please take them out.
.
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