Hello,
I taped a Sensor Fusion Course (one year ago) and I was wondering does
anybody actually know what are the instructor's words? On the tape, it
sounds something like Von Neummann Measure and L2 Measure which can't
be correct. What do you think?
The displayed viewgraphs in class are titled "Ambiguity Function
Scenario" and then "The Ambiguity Function Development."
First, the AMBIGUITY FUNCTION DEFINITION is :
inf
|X(tau, omega)| = | Integral u (zeta) conjugate (u (zeta - tau)) e^(j
omega zeta) d zeta |
inf
Second, the "Ambiguity Function Development" slide goes like this:
Choose u(t) to maximize the measure
E^2 = integral |phi_1 (t) - phi_2 (t)|^2 dt
This can be rewritten as
E^2 = ...
We can maximize the original measure by minimizing:
| integral phi_1 (t) conjugate (phi_2(t)) dt |
This can be rewritten as:
= | ... |
Define the following substitutions
zeta = t - td1
tau = td2 - td1
Third, I think the instructor says the following while looking at the
"Ambiguity Function Development" slide:
And then I use the "L2 Measurement", I look at the integral of the
difference between the two of them squared, but we don't know how to do
that, but there is a theorem, terrible theorm to prove, that the "Von
Neumann Measure" and the "L2 Measure" produce the same results if the
signals are suitably smoothed which these things are, so you can get
about the answer you want by saying I want to come up with u of t, the
one which is the return from the first target minus the integral of phi
two, they will be as different as possible and I can separate out the
targets, through a little bit of mathematics, not bad, because the
absolute value of phi one minus phi two squared is phi one minus phi
two times phi one minus phi two conjugate, I will get this line, it's
not im****tant how I got this line, but you'll observe that this one
doesn't depend on phi two, this one doesn't depend on phi one, so these
two when you realize the only difference between them is where they
exist on the line, these two are equal to each other, this integral and
this integral are equal to each other, so I can't do anything to effect
it, but these over here do admit to the differences between phi one and
phi two and the timing differences, so since this is going to be a
constant, I'm going to minimize what I subtract away, and what I
subtract away in amplitude is the absolute value of phi one minus phi
two conjugate, and if I do a little bit of arithmetic and make some
substittutions, first that tau is the difference in time between them,
not the range to them, that doesn't matter, just the difference in time
between the two of them, and omega which is 2 pi f times the difference
between them in velocity I get the ambiguity function, and this is the
ambiguity function., it is the absolute value of the integral of that
which I'm trying to derive, the original waveform that I should
transmit times the original waveform messed up in every way, conjugated
and tau subtracted away, tau being the differential range, e to the j
omega, omega being 2 pi times the differential velocity, and that tells
me the ambiguity function, and if I minimize the ambiguity function
then I will separate out the two targets in the best possible way. ...
So, what do you think?
Thanks,
Christopher Lusardi
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