"Ray Andraka" wrote in message
news:C7nsf.35562$Mi5.29016@dukeread07...
daestrom wrote:
"Ray Andraka" wrote in message
news:0Gdsf.34358$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.

Actually, I kind of chose those numbers for that very reason ;-)

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.

So what you're saying is, *if* you know the carrier frequency and band-width
of the signal imposed on that carrier, you can design a system that will be
able to reproduce the imposed signal using a relatively low sample rate (low
when compared to the carrier frequency).

But if the carrier frequency changes, then you need to modify the sample
rate to avoid a lot of aliasing issues. So in radio reception, the sample
rate is adjusted along with tuning the receiver? Or is this done at the
intermediate frequency which is fixed so that sample rate adjustment is
fixed with the intermediate frequency? (do they even still use
superheterodyning in tuners?? ;-)

It's been a long time since I did any RF stuff. But A/D and D/A stuff at AF
and lower has been quite a passion for me for some time. And the basic
Nyquist hasn't changed.

daestrom