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Re: [Phys-l] WIFI



On 04/08/2012 11:00 AM, Forinash III, Kyle asked a couple of questions.

Taking the second point first:

This doesn't fit with your comment about this being closer to
frequency modulation so I am confused.

When I say that frequency modulation is essentially the same as
phase modulation, there's no physics in that. It's just math.

In particular: We all know that for a constant, unmodulated
frequency, we can write
V = sin(2 π f t + φ0) [1]
= sin(φ(t))
where the total phase is
φ(t) := 2 π f t + φ0 [2]

Now, if we perform so-called "frequency" modulation, this formula
does *not* generalize to
☠ φ(t) := 2 π f(t) t + φ0 ☠ [3]
because that would be ridiculous at large t.

Instead it generalizes to
φ(t) := 2 π ∫ f(t) dt + φ0 [4]

You can easily verify that equation [4] reduces to equation [2]
when f(t) is a constant.

If we now imagine a carrier frequency f0 (which is constant) plus
some small modulation δf(t), then we obtain:
φ(t) = 2 π f0 t + φ0 + δφ(t) [5]
where
δφ(t) := 2 π ∫ δf(t) dt [6]

which defines the relationship between frequency modulation and
phase modulation. In particular, if the modulation itself is
sinusoidal at frequency Ω then we can do the integral:
δφ(t) = (2 π/Ω) δf(t) [7]

so frequency modulation is just phase modulation in conjunction
with a low-pass filter.

+++ Phase modulation is easier to visualize, so let's stick with
+++ that representation.

One nagging question is how the phase difference is detected. Adding
two identical waves with different phases creates a new wave with a
different amplitude but the frequency and wavelength are the same. So
the phase is read from the amplitude of signal plus reference?

That's mostly the right idea.

a) Actually you want to *multiply* the received signal by the local
oscillator signal. Then filter to pick out the low-frequency part
of the product (throwing away the double-frequency part).

b) Then go one more step down that road. If you're going to generate
a local oscillator signal, you might as well generate *two* of them:
sin(ω t) and
cos(ω t) both.

Then
v1 := lowpass(Vin * sin(ω t))
v2 := lowpass(Vin * cos(ω t))

where v1, v2 are the phasor components of the detected signal. They
directly specify the position of the state on the aforementioned
constellation diagrams:
http://en.wikipedia.org/wiki/Constellation_diagram

==============================

This is relevant to communications ... but it is also *directly*
relevant to physics.

This is a fairly good shibboleth for identifying experimental
physicists: If you ask somebody to measure an AC signal, the
physicist will instinctively reach for a lockin amplifier, in
situations where an ordinary mortal would reach for a voltmeter.

The point is, the lockin provides a local oscillator, and measures
both phasor components, v1 and v2.

True story: Back in the olden days, when computers operated at
a 1MHz clock rate and 1GHz was considered a high frequency, my
colleagues and I built a pulsed-NMR spectrometer that operated
at just over 1GHz. Yeah, I know that's unusually high for NMR.

I hope you're not surprised to hear that we used two mixers, so
as to detect both the in-phase component (v1) and the quadrature
component (v2) of the signal coming in from the antenna. We took
a Fourier transform using v1(t) + i v2(t) as the input ... which
means that positive frequency was definitely different from negative
frequency.

Phase space is two dimensional. If we had measured only v1
(instead of v1 and v2 both) I wouldn't know how to think about
the problem. It's too painful to even think about thinking
about that.

Returning to the communications example: Starting from 256-QAM,
if you throw away the v2 axis in phase space, and collapse
everything down onto the v1 axis, you would be throwing away
something like 15/16ths of the available phase space. Not a
good idea.
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