Kingcosmos said:
Well, from what I read in ADI's Data Conversion Handbook (chap. 2)
what you call 'wanted sidebands' are what they are refering to images
around every multiple of F(s). Let me go read it again for clarity,
maybe I missed something in their explanation.
I would ask Phil to clarify, but he does not have patience for 'stupid
questions' and frankly I do not have patience for assholes who forget
to take their medication. I wish Google Groups had an option to kill-
file.
It helps at this point if you have a background in telecoms; even a Ham
Radio licence would help you. This is one of the difficulties of DSP; you
really need a grounding in quite a number of areas of electronics.
Go back to the old medium wave stations running AM (Amplitude Modulation)
You get the carrier which is the main frequency. Now, even though AM
suggests
that the carrier will change in amplitude, it does not. If the DJ were to
whistle
into the mike at 1kHz, you'll get two sidebands, at frequencies of +/- 1kHz
in addition to the main carrier. The reason for this is that AM is really a
multiplication of the form (1+sin (modulation)) * sin (carrier).
If the DJ speaks into the mike....or.....applies your 500 Hz bandlimited
signal, you'll get one sideband which is a copy of your original spectrum
but shifted up from 0 Hz to the carrier frequency. The other sideband is
an inverted copy of the original spectrum but decreasing downwards
from the carrier.
So, AM is the multiplication of your signal by the carrier frequency.
-----ooooo-----
Come now to sampling. Instead of a carrier frequency of just one sine wave,
you have a series of very short pulses. But, this comes down to (by Fourier
Analysis; another bit of electronics you need under your belt) to a lot of
carriers,
each a multiple of the frequency of the basic sampling rate. Each one of
these
carriers becomes Amplitude Modulated by your original signal. (This time,
sampling is seen as multiplication in the first instance and not as
modulation)
and you get the picture that you described of multiple carriers each with
sidebands
above and below each carrier.
This, I suspect is the picture that you describe as "images". Now, although
this is
a correct use of English, in that each sideband is an image of your original
baseband
signal, it is not what is meant by "imaging" or "image frequencies" when
modulating
one signal by another.
OK, an example off the shelf (and be prepared for some silly errors as I'm
thinking as I'm typing.
Let us suppose that your sampling at 8kHz, which is what you suggested,
and that you've your baseband signal of 500 Hz Bandwidth, but that you've
also got a signal at 7.8kHz, which is crucially above the Fs/2 that everyone
talks about. When we multiply them all by the sampling signal, what do we
get?
(I've deliberately chosen 7.8kHz to guarantee to be able to illustrate
imaging into your wanted 500 Hz bandwidth)
Well first of all, the sampling operations is sin(modulation) * sin(carrier)
and
not (1+sin(modulation)) * sin (carrier) as per the medium wave station, so
we
only get our copies of the sidebands around each multiple of the sampling
frequency. WE DO NOT GET THE UNCHANGING CARRIERS.
Right from the example, (0.5kHz +7.8kHz) * 8kHz will give us 200 Hz
(which is the lower sideband from 7.8kHz), 7.5kHz (which is the lower
sideband
from 0.5kHz), 8.5kHz (which is the upper sideband from 0.5kHz) and 15.8kHz
(which is the upper sideband from 7.8kHz)
Now, it seems at first sight as though there isn't a problem but so far
we've only
considered the first harmonic at 8kHz. We have to consider _ALL_ the
harmonics
including the "zeroth" harmonic of DC, because they're all there in the
sampled
signal.
Lets look at the DC or baseband. Originally we just wanted your 0.5kHz
signal, but now as the result of the modulation at 8kHz, we've also got
a signal at 200Hz resulting from the sampling of the 7.8kHz signal.
This 200Hz signal IS WHAT IS KNOWN AS AN IMAGE.You can't
filter it out once you've got it because it's right in your bandwidth.
OK, now look at what happens at the next harmonic of the sampling
frequency at 16 kHz......you can work out the wanted numbers
from 16+/- 0.5 and 16+/- 7.8.
But.....the 8kHz modulation has created another image at 15.8 kHz,
which appears as an image as a 200 Hz from (16 - 15.8) in what
should be the lower sideband of the 16 kHz carrier.
And so it goes on for every subsequent harmonic of the original sampling
frequency.
Right, I'm getting bored with this, it's very tedious! There's one final
point
to make and it is to introduce where the Nyquist criterion comes from.
Consider two of the carrier, say the 8kHz and 16kHz already discussed.
Provided no upper sideband of the 8kHz carrier is above 12 kHz, and
provided no lower sideband of the 16kHz carrier is below 12 kHz, then
there won't be any aliasing. ie, the bandwidth of the sidebands has to
be less than the distance between two carriers, or, put another way,
the maximum frequency in the baseband must not exceed half
the sampling frequency.....as I said, you need to have a bit
of a background in radio.
(I am a Radio Ham with the callsign G....4....S....D....W)