Re: Nyquist????

On Mon, 27 Oct 2008 01:23:38 -0800, glen herrmannsfeldt
<gah@xxxxxxxxxxxxxxxx> wrote:

Also, consider sampling of AM-DSB signals.

But the sidebands come from 2sin(a)sin(b)=cos(a-b)-cos(a+b)

they are created in the modulation, not moved up from
positive and negative frequencies.

I'm not so sure I'd agree with that. Let's use cosine modulation
instead of sine, because cos(0)=1 ...

cos(a)cos(b) = ½[cos(a-b)+cos(a+b)]

.... and pretend that a baseband signal is modulated by DC. Assume
that the signal being modulated is composed of several sines and
cosines of various frequencies:

cos(a)[cos(b)+sin(c)+...+cos(y)+sin(z)] =
cos(0)[cos(b)+sin(c)+...+cos(y)+sin(z)] =
cos(b)+sin(c)+...+cos(y)+sin(z) [the original signal]

Now let's assume that the signal is composed of several cosines and
sines of various frequencies, all of which are GREATER than the
carrier frequency (associated with "cos(a)"):

cos(a)[cos(b)+sin(c)+...+cos(y)+sin(z)] =
cos(a)cos(b)+cos(a)sin(c)+...+cos(a)cos(y)+cos(a)sin(z) =
½[cos(a-b)+cos(a+b)] +
½[sin(a+c)-sin(a-c)] +
... +
½[cos(a-y)+cos(a+y)] +
½[sin(a+z)-sin(a-z)]

In this case the lower sideband of the modulated positive frequencies
falls at negative frequencies, and the upper sideband of the modulated
negative frequencies falls at positive frequencies. We might consider
this to be analogous to aliasing, since the envelopes of the spectra
overlap.

Now let's assume that the signal is composed of several cosines and
sines of various frequencies, all of which are LESS than the carrier
frequency (associated with "cos(a)"). Again:

cos(a)[cos(b)+sin(c)+...+cos(y)+sin(z)] =
cos(a)cos(b)+cos(a)sin(c)+...+cos(a)cos(y)+cos(a)sin(z) =
½[cos(a-b)+cos(a+b)] +
½[sin(a+c)-sin(a-c)] +
... +
½[cos(a-y)+cos(a+y)] +
½[sin(a+z)-sin(a-z)]

In this case, however, there is no overlap between the lower sideband
of the modulated positive frequencies and the upper sideband of the
negative modulated frequencies.

In other words, there is no loss of generality in viewing a baseband
signal as a two-sided signal modulated by DC.
.

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