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Old July 22nd 05, 01:10 AM
K7ITM
 
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Assuming that you have a bandpass filter that allows your desired band
to pass (say 43.5 to 44.5MHz), and that has enough attenuation outside
some band (say 42.5 to 45.5MHz) to protect against aliases, then the
only harmonics of the sampling frequency that are of interest are the
first one below the passband, and the first one above the passband.
That is assuming that you don't have a harmonic of the sampling
frequency inside the passband, which I explained would be bad because
you can't distinguish between a signal xx.xxkHz above that harmonic and
a signal xx.xxkHz below that harmonic.

You can list some specific rules. Try this:

Define four frequencies for your filter:
f1 = frequency below which attenuation is great
enough that you can ignore signals there.
f2 = bottom edge of the passband, where
attenuation is small enough that signals are OK.
f3 = top edge of the passband, like f2, so signals
between f2 and f3 are all OK
f4 = frequency above which attenuation is great
enough that you can ignore signals there.
So, f1f2f3f4.
(in the example above, f1=42.5MHz, f2=43.5MHz, f3=44.5MHz and
f4=45.5MHz. But the passband does not have to be symmetrical; f4 could
have been 46.5MHz for example, and the equations below will still
work.)
Define the sampling frequency = fs
Then there are two possibilities that will work:
(1) a harmonic of fs, call it n*fs, falls below (f1+f2)/2,
and (n+1)*fs-f4 f3-n*fs. In that case, the band
from f2 to f3 will be aliased down to the band
from f2-n*fs to f3-n*fs. In fact, each frequency
f in the passband will alias (mix) to f-n*fs.
OR
(2) (kind of the mirror image of (1)) a harmonic of
fs, call it m*fs, falls above (f3+f4)/2, and
m*fs-f2 f1-(m-1)*fs. In that case, each
frequency f in the passband will alias to m*fs-f.
In other words, the output spectrum will be
flipped compared with the input: lower freqs
in the input spectrum will be at the higher end
of the output (digitized) spectrum.

In both (1) and (2), the first requirement as I've stated them is so
that nothing on the other side of the sampling harmonic from the
passband is big enough to cause trouble in the output frequency range.
The second requirement insures that nothing aliased by the harmonic on
the other side of the passband from the harmonic doing the work will
cause trouble in the output passband. But bewa the digitized
output may very well contain frequencies which lie outside the
passband, because your analog filter on the passband does not cut them
off sharply enough to get rid of them. You can use DSP algorithms (FIR
or IIR filters, for example) to do a good job getting rid of the
remainder that you might not want in the output.

Cheers,
Tom

(Sure hope I got all those inequalities right! I trust someone will
double-check them and let us know if they are wrong...)