mixing a sinewave with time delayed replica of itself

Hello,
The mathematics of frequency mixing seems so simple, but I always end up
getting confused when I am trying to relate the math to what I observe in
the lab - I don't get the correct products - have to change a sign or
something to get things to work out correctly. Right now, I am trying to
come up with the power transfer function of sinewave that is delayed and
frequency, mixed with itself.  One way I think of an "ideal mixer" in the
time domain is as a multipler - in the frequency domain I think of it as a
convolver.  But if I think of the mixing process as a difference in the
frequency domain (which seems intuitively obvious), I get the power
transfer function that corresponds with my observations: H(w)=1-e^jwT, or 
|H(w)|^2=2*(1-cos(w*T)), where T is the delay.  I tried to prove to myself
that a mixer performs a difference in the frequency domain by taking the
simplest case, but couldn't work it out.  If I difference the fourier
transform of two complex functions, e^jw1t and e^jw2t (to inspect the
resulting difference frequency w1-w2) I have to difference two delta
functions, and, embarassingly I don't know how to do that.  I then got
even more confused when I tried to just sit down and prove to myself that
the mixer does indeed perform a convolution, by convolving the
aforementioned delta functions, but ended up with a delta function at
w1+w2 - not w1-w2.    Can someone help me out with this?  Is there a paper
or book that lines out frequency mixing so I can understand the basic math
of mixing? I would assume this is basic linear systems stuff. And is it
valid to think of mixing as a difference, not a convolution, in the
frequency domain?  
Thanks,
John M.        

JohnReno wrote:
> Hello,
> The mathematics of frequency mixing seems so simple, but I always end up
> getting confused when I am trying to relate the math to what I observe in
> the lab -

Mixing a sine wave with itself will give you a sine with 2x the
frequency and a DC term.  The value of the DC term depend on the PHASE
relationship betwen the two inputs to the mixer...

A mixer and phase detector are both fundamentally multipliers.

When used as a phase detector, the low frequency DC term is kept and
the higher stuff is filtered out. 

Mark