On 01/08/2011 09:02 PM, dvsarwate wrote:
> On Jan 8, 5:05 pm, Randy Yates<ya...@ieee.org> wrote:
>
>> I asked what kind of modulator there could be, other than a linear modulator,
>> for QAM. Now maybe I missed it (I seem to be missing a lot lately...), but I
>> don't see an answer to that question here.
>
> All right, regardless of whatever else anyone may have told you, QAM
> **is not** a linear modulation scheme. (See Eric Jacobsen's comments
> re this matter too). There are 4 inputs to a 16-QAM transmitter with
> each input being restricted to have value 0 or 1, and the output is
> *not* a linear function of these 4 inputs, whether you take them
> singly or in pairs or collectively.
What is your definition of "linear modulation?" One definition
that could be used is from [vantrees, p.467]:
The modulated signal is s(t, a(t)). Denote the component of s(t, a(t))
that does not depend on a(t) as c_o(t). If the signal [s(t, a(t)) - c_o(t)]
obeys superposition, then s(t, a(t)) is a linear modulation system.
So, under that definition, I claim that QAM is a linear modulation
scheme as follows.
Let the nth input to a 16-QAM modulator be
a(t) = (p[n(t)], q[n(t)], r[n(t)], and s[n(t)]),
each of which is from the set of two integers (0, 1), where n denotes
the symbol index. Let the output of the modulator s(t, a(t)) be denoted as
s(t, a(t)) = x(t) + i*y(t),
where i = sqrt{-1} and where
x(t) = (-3 + 2*A[n(t)]) * v(t - n(t)*T),
A[n(t)] = 2*p[n(t)] + q[n(t)],
v(t) = pulse shape,
T = symbol period,
n(t) = floor(t/T)
Similarly
y(t) = (-3 + 2*B[n(t)]) * v(t - m*T),
B[n(t)] = 2*r[n(t)] + s[n(t)]
Now we can rewrite this as
s(t, a(t)) = c_a(t) + c_o(t)
where c_a(t) is the part of the signal that depends on the input a(t),
c_a(t) = 2*A[n(t)]*v(t - n(t)*T) + i*2*B[n(t)]*v(t - n(t)*T)
and c_o(t) is the "other" part
c_o(t) = [-3 *v(t - n(t)*T)] * (1 + i).
Then since c_a(t) obeys the superposition principle, 16-QAM is a linear
modulator.
> If you nevertheless insist
> that 16-QAM is a linear modulation scheme, then please
> tell us what the output signal is for inputs
>
> (1,1,0,0)
>
> (0,0,1,1)
>
> (1,1,1,1)
>
> Now, prove that the third output of the three that I have
> asked you to provide is the sum of the first two outputs
> that you have provided us. After all, a linear scheme
> -- call it L for convenience -- should have the property
> L(x+y) = L(x) + L(y), no?
No, not by the definition above. I think that's the problem with this
"linear modulation" thing - it's not well defined across the
profession. I actually was initially thinking of Proakis' definition
[proakiscomm], but found it his "definition" was weak.
>> Do you know where I can find some of Jesus' spit?
>
> Huh?
A reference to the blind being healed: [bible, Mark 8:23]
>
> --Dilip Sarwate
>
--Randy
@book{vantrees,
title = "Detection, Estimation, and Modulation Theory, Part I",
author = "Harry L. Van Trees",
publisher = "Wiley",
year = "2001"}
@BOOK{proakiscomm,
title = "{Digital Communications}",
author = "John~G.~Proakis",
publisher = "McGraw-Hill",
edition = "fourth",
year = "2001"}
@book{bible,
title = "Bible},
author = "God",
year = "all"}
--
Randy Yates % "My Shangri-la has gone away, fading like
Digital Signal Labs % the Beatles on 'Hey Jude'"
yates@digitalsignallabs.com %
http://www.digitalsignallabs.com % 'Shangri-La', *A New World Record*, ELO