Hi Lawrence,

Very good explanation and I'm glad to see that you have introduced Lile's
Law ;-)

One little correction, though: to be able to recover the information from a
signal, it is NOT necessary to sample at a rate which is more than twice
the maximum frequency content. It is only necessary to sample at a rate
which is more than twice the bandwidth. So, for example, if you have a
signal at 10MHz which is amplitude modulated by a 10kHz signal, the
spectrum looks like a narrow band (20kHz wide) way up at 10MHz and nothing
else. The signal bandwidth in this case is actually 10kHz (half the width
of the band of freqs that you see) and you need only sample at 20kHz (or
preferably a little higher to prevent problems, as you say). You DO,
though, need an ADC which has a very small sample aperture (i.e., it's
sample and hold just looks at a very small region in time to grab each sample).

This makes it easier to do things like software radio where you do
demodulation and IF filtering in DSP. Your IF may be at RF frequencies, but
you need only sample according to the bandwidth.

Sean

At 10:44 AM 2/28/02 -0600, Lawrence Lile wrote:
>Let's try plainer plain Inglitsch:
>
>A low pass filter will allow low frequencies through without much loss, will
>tend to block high frequencies, and at some magic frequency determined by
>the values of the components will attenuate the signal by about half.
>Frequencies above this are attentuated more, below this are attenuated less.
>
>
>A classic low pass filter (there are dozens of different types) is simply:
>
>Vin--------R--------OUT
>                     |
>                     C
>                     |
>                     GND
>
>This is useful in PICs to take a high frequency signal, say a PWM output,
>and turn it into a smooth nearly DC signal, say for an input to a
>comparator.
>
>Calculus is handy for figgering out how this stuff really works, but it is
>possible to understand it in a practical way without it.  I was building
>stuff with low pass filters long before I ever studied calculus.
>
>
>OK, what about sampled data?  An old fella named Nyquist said that you need
>to sample your data at least twice the highest frequency present in the
>signal for it to make any sense.  Another fella named Lile said that Nyquist
>was an optimist.  A low pass filter is often used in front of a data
>acquisition system of any kind to limit high frequencies.  If the high
>frequency cutoff of your filter is 10KHZ, you need to sample at least 20KHZ
>(per Nyquist) and realistically even faster (per Pessimistic Mr. Lile) to
>get good useful data without aliasing and other problems.
>
>Next you are going to ask how to calculate the cutoff frequency of such a
>filter.  I'll bet there are some good cookbook methods that don't involve
>any calculus.  I, being lazy, just chuck parts into BSpice and let it figger
>the answer for me.  Anybody want to weigh in?
>
>--Lawrence Lile

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