Hello,
I taped a Sensor Fusion Course (one year ago) and I was wondering does
anybody actually know what are the words the instructor is using? 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 some background, the AMBIGUITY FUNCTION DEFINITION is :
inf
|X(tau, omega)| = | Integral u (zeta) conjugate (u (zeta - tau)) e^(j
omega zeta) d zeta |
inf
Now, 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 important 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
[Ambiguity Function Derivation] What Is professor's Words (Von Neumann Measure? / L2 Measure?)
Started by ●May 8, 2006
Reply by ●May 8, 20062006-05-08
<clusardi2k@aol.com> wrote in message news:1147110004.986104.64020@g10g2000cwb.googlegroups.com...> Hello, > > I taped a Sensor Fusion Course (one year ago) and I was wondering does > anybody actually know what are the words the instructor is using? On > the tape, it > sounds something like Von Neummann Measure and L2 Measure which can't > be correct. What do you think? >a lot of stuff snipped> > Thanks, > Christopher Lusardi >Hello Christopher, In quantum mechanics one talks of a von Neumann measure that yields the degree of entanglement between states in a composite system. So in terms of an ambiguity, that makes sense. And L2 measure is probably a Lebesque measure using L2 norms. IHTH, Clay
Reply by ●May 10, 20062006-05-10
Clay S. Turner wrote:> <clusardi2k@aol.com> wrote in message > news:1147110004.986104.64020@g10g2000cwb.googlegroups.com... > > Hello, > > > > I taped a Sensor Fusion Course (one year ago) and I was wondering does > > anybody actually know what are the words the instructor is using? On > > the tape, it > > sounds something like Von Neummann Measure and L2 Measure which can't > > be correct. What do you think? > > > > a lot of stuff snippedMore stuff snipped> And L2 measure is probably a Lebesque > measure using L2 norms.For most practical purposes (at least what DSP is concerned), that translates to plain English as "the square root of the integrated squared amplitude", or even simpler, "RMS". BTW, I once herad that "LA" (the name of that Californian city) probably is *the* most efficient acronym anywhere? The original name was, apparently, a *lot* longer than just "Los Angeles"? Rune
Reply by ●May 10, 20062006-05-10
Clay S. Turner wrote:> <clusardi2k@aol.com> wrote in message > news:1147110004.986104.64020@g10g2000cwb.googlegroups.com... > > Hello, > > > > I taped a Sensor Fusion Course (one year ago) and I was wondering does > > anybody actually know what are the words the instructor is using? On > > the tape, it > > sounds something like Von Neummann Measure and L2 Measure which can't > > be correct. What do you think? > > > > a lot of stuff snipped > > > > > Thanks, > > Christopher Lusardi > > > > Hello Christopher, > > In quantum mechanics one talks of a von Neumann measure that yields the > degree of entanglement between states in a composite system.I'v never heard of a "von Neumann measure" in any course on measure theory or quantum information theory. I've heard of the "von Neumann entropy", which is a measure of several things, one of which is the degree of entanglement of bipartite pure states. Not having read Christopher's transcript, I can't really tell what is meant by the word "measure" - is this a quantum mechanics course or a measure theory course or something else? Also, there is no such thing as an L2 measure. L2 is a family of functions on a measurable space. If one choses the Lebesgue measure we get the familiar space of bounded energy signals. "Counting" measure -> the space of square summable discrete signals. etc. Regards, Andor
Reply by ●May 10, 20062006-05-10
Rune Allnor wrote:> Clay S. Turner wrote: > >><clusardi2k@aol.com> wrote in message >>news:1147110004.986104.64020@g10g2000cwb.googlegroups.com... >> >>>Hello, >>> >>>I taped a Sensor Fusion Course (one year ago) and I was wondering does >>>anybody actually know what are the words the instructor is using? On >>>the tape, it >>>sounds something like Von Neummann Measure and L2 Measure which can't >>>be correct. What do you think? >>> >> >>a lot of stuff snipped > > > More stuff snipped > > >>And L2 measure is probably a Lebesque >>measure using L2 norms. > > > For most practical purposes (at least what DSP is concerned), that > translates to plain English as "the square root of the integrated > squared amplitude", or even simpler, "RMS". > > BTW, I once herad that "LA" (the name of that Californian city) > probably > is *the* most efficient acronym anywhere? The original name was, > apparently, a *lot* longer than just "Los Angeles"?El Pueblo de Nuestra Se�ora Reina de los �ngeles de la Porciuncula. (The Town of Our Lady Queen of the Angels of the pig sty?) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 10, 20062006-05-10
Comments below: "Andor" <andor.bariska@gmail.com> wrote in message news:1147251600.259361.163190@u72g2000cwu.googlegroups.com...> Clay S. Turner wrote: >> <clusardi2k@aol.com> wrote in message >> news:1147110004.986104.64020@g10g2000cwb.googlegroups.com... >> > Hello, >> > >> > I taped a Sensor Fusion Course (one year ago) and I was wondering does >> > anybody actually know what are the words the instructor is using? On >> > the tape, it >> > sounds something like Von Neummann Measure and L2 Measure which can't >> > be correct. What do you think? >> > >> >> a lot of stuff snipped >> >> > >> > Thanks, >> > Christopher Lusardi >> > >> >> Hello Christopher, >> >> In quantum mechanics one talks of a von Neumann measure that yields the >> degree of entanglement between states in a composite system. > > I'v never heard of a "von Neumann measure" in any course on measure > theory or quantum information theory. I've heard of the "von Neumann > entropy", which is a measure of several things, one of which is the > degree of entanglement of bipartite pure states.I've seen in physics papers where one defines "von Neumann entropy" and then later refers to it by "von Neumann measure."> > Not having read Christopher's transcript, I can't really tell what is > meant by the word "measure" - is this a quantum mechanics course or a > measure theory course or something else? > > Also, there is no such thing as an L2 measure. L2 is a family of > functions on a measurable space. If one choses the Lebesgue measure we > get the familiar space of bounded energy signals. "Counting" measure -> > the space of square summable discrete signals. etc.The term "L2 measure" is coming into usage. Just google for it to see some examples. I think that these terms might have been thrown around sloppily - but one would hope that the course lectures provided the necessary framework. I only attempted to show the OP, that the terms are real terms (even if used rather sloppily). They may not apply to his lecture material - but he can't just ignore these terms without more information. I like the term Lebesgue measure and I linked it to the use of "L2 measure" to increase the OP's space of places to look. For details on Lebesque measure, I just look in my copy of "Measure and the Integral" by Henri Lebesgue. I inherited a lot of old books from my dad. Clay> > Regards, > Andor >
Reply by ●May 10, 20062006-05-10
Clay S. Turner wrote:> >> In quantum mechanics one talks of a von Neumann measure that yields the > >> degree of entanglement between states in a composite system. > > > > I'v never heard of a "von Neumann measure" in any course on measure > > theory or quantum information theory. I've heard of the "von Neumann > > entropy", which is a measure of several things, one of which is the > > degree of entanglement of bipartite pure states. > > I've seen in physics papers where one defines "von Neumann entropy" and then > later refers to it by "von Neumann measure."Ok. Just because I've never heard of it doesn't mean anything :-).> > > > > Not having read Christopher's transcript, I can't really tell what is > > meant by the word "measure" - is this a quantum mechanics course or a > > measure theory course or something else? > > > > Also, there is no such thing as an L2 measure. L2 is a family of > > functions on a measurable space. If one choses the Lebesgue measure we > > get the familiar space of bounded energy signals. "Counting" measure -> > > the space of square summable discrete signals. etc. > > > The term "L2 measure" is coming into usage. Just google for it to see some > examples.The last link on the first page for googling "L2 measure" seems to be the most relevant: http://mathworld.wolfram.com/L2-Space.html> > I think that these terms might have been thrown around sloppily - but one > would hope that the course lectures provided the necessary framework. I only > attempted to show the OP, that the terms are real terms (even if used rather > sloppily). They may not apply to his lecture material > - but he can't just ignore these terms without more information. > > I like the term Lebesgue measure and I linked it to the use of "L2 measure" > to increase the OP's space of places to look. For details on Lebesque > measure, I just look in my copy of "Measure and the Integral" by Henri > Lebesgue. I inherited a lot of old books from my dad.You inherited Lebesgue's "Measure and the Integral" from your dad? (bowing my head) What did your dad do for a living (if you don't mind) .. ?
Reply by ●May 10, 20062006-05-10
"Andor" <andor.bariska@gmail.com> wrote in message news:1147279478.554847.208310@j33g2000cwa.googlegroups.com...> > You inherited Lebesgue's "Measure and the Integral" from your dad? > > (bowing my head) > > What did your dad do for a living (if you don't mind) .. ? >Andor, He was a professor of Mathematics (Biostatisitics). I have books by Cramer, Rao, Pearson among others. My dad liked to collect old books. I have a 1930's physics text by Millikan. Plus I have a copy of Fermi's notes bound into a book. I even have a copy of Gauss's work (modern translation) on the theory of least squares! Another on probability by Laplace. And more! I'm the only one of his 8 kids who is into math, so I got all of the technical books. It will take years to read all of them; however, it will be fun. Clay
Reply by ●May 11, 20062006-05-11
Clay S. Turner wrote:> "Andor" <andor.bariska@gmail.com> wrote in message > news:1147279478.554847.208310@j33g2000cwa.googlegroups.com... > > What did your dad do for a living (if you don't mind) .. ?> He was a professor of Mathematics (Biostatisitics). I have books by Cramer, > Rao, Pearson among others. My dad liked to collect old books. I have a > 1930's physics text by Millikan. Plus I have a copy of Fermi's notes bound > into a book. I even have a copy of Gauss's work (modern translation) on the > theory of least squares! Another on probability by Laplace. And more! I'm > the only one of his 8 kids who is into math, so I got all of the technical > books. It will take years to read all of them; however, it will be fun. > > ClayWhat do you do to maintain them? Here's my throw-away tip of the year. IYou can get cheap (because they are international printings) books through www.gettextbooks.com Thanks everyone, Christopher Lusardi
