How much are the carrier frequency higher than the baseband signal frequency for a band-limited AWGN channel?

When it comes to basic modulation method, ASK/FSK/PSK, there is a
sentence in some textbooks: the carrier frequency should be far more
than the baseband signal frequency. However, no one give the detail. I
need to analyze the data rate upper limit of a communication system
using QPSK coherent demodulation in a band-limited AWGN channel when
the carrier frequency and channel bandwidth is known. Shannon-Hartley
Capacity Theorem give some information, but without the carrier
frequency.

On Dec 19, 9:31 am, "X.Y." <Xieyu1...@gmail.com> wrote:
> When it comes to basic modulation method, ASK/FSK/PSK, there is a
> sentence in some textbooks: the carrier frequency should be far more
> than the baseband signal frequency. However, no one give the detail. I
> need to analyze the data rate upper limit of a communication system
> using QPSK coherent demodulation in a band-limited AWGN channel when
> the carrier frequency and channel bandwidth is known. Shannon-Hartley
> Capacity Theorem give some information, but without the carrier
> frequency.

This is a difficult question, at least in the way you posed it.

Coarsely speaking, the carrier frequency should at least be
sufficiently
high such that carrier modulation can translate a complex-valued
baseband signal into a real-valued baseband signal.  I am assuming
that your channel is real-valued, correct?

Then using ideal sinc filters the answer is fc>R the symbol rate.
But in practice you have to use something like raised cosine pulse
shaping, and truncated lowpass filters for image rejection in
demodulation.  So in this sense you will "lose" some performance
if you do symbol-by-symbol detection per the usual textbook coherent
QPSK demodulation technique.  How much you lose is determined by
how sharp you can make your filters in the frequency domain.

Hope this helps.
Julius

On Dec 20, 12:44&#4294967295;am, julius <juli...@gmail.com> wrote:
> On Dec 19, 9:31 am, "X.Y." <Xieyu1...@gmail.com> wrote:
>
> > When it comes to basic modulation method, ASK/FSK/PSK, there is a
> > sentence in some textbooks: the carrier frequency should be far more
> > than the baseband signal frequency. However, no one give the detail. I
> > need to analyze the data rate upper limit of a communication system
> > using QPSK coherent demodulation in a band-limited AWGN channel when
> > the carrier frequency and channel bandwidth is known. Shannon-Hartley
> > Capacity Theorem give some information, but without the carrier
> > frequency.
>
> This is a difficult question, at least in the way you posed it.
>
> Coarsely speaking, the carrier frequency should at least be
> sufficiently
> high such that carrier modulation can translate a complex-valued
> baseband signal into a real-valued baseband signal. &#4294967295;I am assuming
> that your channel is real-valued, correct?
>
> Then using ideal sinc filters the answer is fc>R the symbol rate.
> But in practice you have to use something like raised cosine pulse
> shaping, and truncated lowpass filters for image rejection in
> demodulation. &#4294967295;So in this sense you will "lose" some performance
> if you do symbol-by-symbol detection per the usual textbook coherent
> QPSK demodulation technique. &#4294967295;How much you lose is determined by
> how sharp you can make your filters in the frequency domain.
>
> Hope this helps.
> Julius

Hi,Julius, Thanks for your reply.

what you means is to analyze this question from the point of filter,
am I right? The transmitter, channel and receiver can all be seen as
filters. As you mentioned, &#4294967295; How much you lose is determined by how
sharp you can make your filters in the frequency domain.&#4294967295; In this
case, &#4294967295;lose&#4294967295; means lose of frequency components. Correspondingly, a
square wave becomes a sine wave, right? In the channel of my system,
there is a LC tank circuit, and the carrier frequency is the same as
the LC resonance frequency. So these filters just influent baseband
signal. The question is how much the baseband signal is harmed,
especially its shaped is changed, that it cannot be coherent
demodulated?

On Dec 22, 1:31 am, "X.Y." <Xieyu1...@gmail.com> wrote:
> On Dec 20, 12:44 am, julius <juli...@gmail.com> wrote:
>
>
>
> > On Dec 19, 9:31 am, "X.Y." <Xieyu1...@gmail.com> wrote:
>
> > > When it comes to basic modulation method, ASK/FSK/PSK, there is a
> > > sentence in some textbooks: the carrier frequency should be far more
> > > than the baseband signal frequency. However, no one give the detail. I
> > > need to analyze the data rate upper limit of a communication system
> > > using QPSK coherent demodulation in a band-limited AWGN channel when
> > > the carrier frequency and channel bandwidth is known. Shannon-Hartley
> > > Capacity Theorem give some information, but without the carrier
> > > frequency.
>
> > This is a difficult question, at least in the way you posed it.
>
> > Coarsely speaking, the carrier frequency should at least be
> > sufficiently
> > high such that carrier modulation can translate a complex-valued
> > baseband signal into a real-valued baseband signal.  I am assuming
> > that your channel is real-valued, correct?
>
> > Then using ideal sinc filters the answer is fc>R the symbol rate.
> > But in practice you have to use something like raised cosine pulse
> > shaping, and truncated lowpass filters for image rejection in
> > demodulation.  So in this sense you will "lose" some performance
> > if you do symbol-by-symbol detection per the usual textbook coherent
> > QPSK demodulation technique.  How much you lose is determined by
> > how sharp you can make your filters in the frequency domain.
>
> > Hope this helps.
> > Julius
>
> Hi,Julius, Thanks for your reply.
>
> what you means is to analyze this question from the point of filter,
> am I right? The transmitter, channel and receiver can all be seen as
> filters. As you mentioned, &#4294967295; How much you lose is determined by how
> sharp you can make your filters in the frequency domain.&#4294967295; In this
> case, &#4294967295;lose&#4294967295; means lose of frequency components.

> Correspondingly, a
> square wave becomes a sine wave, right?

I don't understand what you said here...  square wave in what domain?

> In the channel of my system,
> there is a LC tank circuit, and the carrier frequency is the same as
> the LC resonance frequency. So these filters just influent baseband
> signal.

No.  When you convert from passband to baseband you have to
cancel the carrier images at 2fc.  So unless you have ideal
lowpass filters, these images will have aliasing into the baseband
signal, too.

> The question is how much the baseband signal is harmed,
> especially its shaped is changed, that it cannot be coherent
> demodulated?

Here you seem to be asking a question about inter-symbol
interference, or equivalently of distortion of your signal shape
due to a linear, time-invariant filter that is your "channel".

This can be derived simply by sending a single symbol through
your passband channel, and using matched filtering at the
output.  Assuming that your pulse shape satisfies the Nyquist
ISI criterion, then you can evaluate the output of the matched
filters at integer multiples of the symbol rate, and you can think
of this as your "effective" baseband channel.

It seems to me that you are confusing a bunch of things at
once, you should realize that I've mentioned a few issues here:

1.  Effect of non-ideal lowpass filters when converting from
     passband to baseband.  This effect is NOT linear and
     time-invariant.  In most cases you can model this effect
     as additive white Gaussian noise.  If you want to be more
     accurate then you have to better characterize your
     transmitted signal, since this is the only source of
     randomness of this "noise".

2.  Effect of the passband channel viewed by the baseband
     modulator/demodulator.  This effect CAN be modeled as
     linear and time-invariant.

Hope this helps.
Julius