daestrom wrote:
"Ray Andraka" wrote in message
news:0Gdsf.34358$Mi5.34121@dukeread07...

daestrom wrote:

So, if I have a signal with a 1000 hz carrier, with a bandwidth of 50 hz,
you think I can sample it at just 150 hz and get accurate reproduction?
That's just wrong.

It is the maximum frequency component in the signal that is important.
The bandwidth is not related unless the lower edge of the band is at 0 hz
(whereupon the upper side of the band is equal to the max frequency).

daestrom



No, it is correct. If you have a signal with a 1000 Hz carrier and a 50
Hz Bandwidth, you can indeed sample it at 150 Hz and get accurate
reproduction...provided the rest of the spectrum is clear. That
requirement is typically provided with an anti-alias filter. In this
case, the anti-alias filter has to be a BAND-PASS filter centered on 1000
Hz rather than the low pass filter associated with baseband sampling.
This works because sampling folds the spectrum (aliasing) so that parts of
the frequency band with higher frequency than the sampling frequency get
folded back onto baseband. As long as the full spectrum only has energy
in a bandwidth less than or equal to half the sample rate, you get all of
the information necessary to reconstruct the original signal (assuming you
know the characteristics of the fixed anti-alias filter so that you know
which image to select when you unfold the spectrum). If there was signal
energy outside of the Fs/2 bandwidth, it adds to signal inside the
bandwidth during the folding that sampling causes, and then you lose
information since there is no way to separate the energy if it has been
added with other energy by folding.


You are in effect demodulating the incoming signal and sampling the result,
not sampling the incoming signal. You are 'throwing away' the information
that would tell you what the carrier freq is.

Now, in radio that may be all well and good, since demodulation is a
necessary part of reception anyway. But some of us were talking about
reproducing the incoming signal, not stripping out the low freq component of
some carrier.

Note that if the carrier is an exact multiple of the sample rate, *then* an
unmodulated carrier will produce no 'alias' signal. But 150 doesn't go
evenly into 1000.

If you have a completely unmodulated 1000 hz signal, passed through a 50 hz
wide band-pass, centered around 1000 hz and sampled at 150 hz, your sampled
data is indistinguisable from that of a 25 hz signal. Even knowing the
band-pass filter's characteristic doesn't tell me if the carrier was
unmodulated 1000 hz, or if there was a true 30 hz signal modulating it.

daestrom



The information that tells you the frequency of the carrier is not
discarded, but is partially implied by the system, just as it is with a
baseband system. Remember, sampling is essentially the mixing of the
signal with an impulse train, followed by a sample rate decimation
without any filtering. The choice of frequencies in this example are
unfortunate because there is in fact some interference between the
positive and negative frequency images of the original signal. When
dealing with real-only inputs, you need to be judicious in selecting the
sample frequency so that the frequency folding does not fold the
negative image (that is a reflection of the positive image and is always
present for a real signal) onto the positive image. Still, that doesn't
mean that the sample frequency has to be a sub-multiple of the carrier.
For example, 160 Hz sampling works (as does 210 Hz) with a 1000Hz
signal that has a 50Hz bandwidth because it puts both the positive and
negative frequency images into the sampled spectrum without overlap.
There is sufficient information there to reconstruct the original signal
if the center frequency of the anti-alias filter is known.

And yes, you are correct that the sampled signal is indistinguishable
from one which it aliases onto: but those other frequencies are not
present in the signal thanks to the anti-alias filter. The point is
that the anti-alias filter needn't be a low pass filter. It can be a
band pass filter as long as the bandwidth is less than half the sample
frequency. If the input signal is a real signal, there are additional
considerations to make sure that the postive and negative frequeyncy
images do not overlap when the spectrum is folded.