> Hi Julius,
> Nice to hear from you. Hope ya' know
> that we wish you all the best.
>
> See Ya'
> [-Rick-]
Rick, I saw that several engineers here have a copy of your book, and
I get a big kick out of showing that my name appears in the
acknowledgements section of your book!
Julius
Reply by kl31n●December 11, 20062006-12-11
Il 11 Dec 2006 11:16:16 -0800, julius ha scritto:
> kl31n wrote:
>> I couldn't find the discussion on the accuracy of the results depending on
>> how many samples you use of the input signal. That's the part I'm
>> interested in actually. As I said already maybe I'm not seeing the obvious,
>> but would anybody be so kind to point me to a reference where not the
>> problem is solved but it's solved with finite data sets and the accuracy of
>> the results is discussed together with the general approach?
>>
>> Thank you again,
>>
>> kl31n
> OK, I think I see where your hang-up is. I think the easiest way to
> get the answer is to write out your observation in terms of signal + noise,
> for example:
>
> y_n = A x_n + z_n. ... (1)
>
> x_n will be your cos() and sin() in vector form, z_n is your noise, and
> A
> will be the vector of your wanted parameters.
>
> In your original post you weren't clear whether the AWGN is in
> continuous-time domain or in discrete-time domain. It will make a
> big difference in how the variance of z_n will scale with T your
> sampling
> period. So you should clarify your model first.
First of all thanks for your answer. The noise is in the discrete-time
domain.
> Now, once you obtain (1) you can then apply your least-squares fit,
> essentially by writing:
>
> y = X a + z,
>
> and obtaining the estimate \hat{a} from the pinv of X and the
> observation y. This is called the Best Linear Unbiased Estimate, and I
> would be very surprised if an estimation theory book does not cover
> this example.
I've been able to find the solution to such a point in the Kay's book you
suggested.
In particular it can be shown that the solution calculated by the
pseudo-inverse provides MVU estimates \hat{a} for the parameters of the
model you wrote and its covariance matrix is known to be \sigma^2 * (X' *
X)^ -1, provided the covariance of the noise vector is \sigma^2 * I.
For the X we're talking about it's possible to see that (X'*X) = (M/2) * I,
where M is the number of rows in X.
Now the problem however is that my results differ from the ones that
Vladmir suggested as while he suggested that accuracy would be sqrt(M)
times better, it results to me that it's sqrt(M/2) better...I think I
didn't forget anything(Vladimir if you're still reading this post...did I
misunderstand you somehow?).
> Realize that X is completely known because you know what the frequency
> is, so now you can also do a little experiment to see if the quality of
> the estimate of a depends on the frequency or not.
Well, correct me if I'm wrong, but there's no actual need to test such a
thing. A MVU provides CRLB results uniformly with the actual values of the
parameters and you can consider any \deltaf right in the parameters
themselves.
> Hope this helps.
Thanks for you time!
> Julius
kl31n
Reply by Rick Lyons●December 11, 20062006-12-11
On 11 Dec 2006 12:19:09 -0800, "julius" <juliusk@gmail.com> wrote:
>
>Randy Yates wrote:
>
>> Good to see you here, Julius. Your presence boosts the groups
>> IQ a couple of decades!
>>
>> Schlumberger - aren't they essentially what used to be National
>> Semiconductor?
>
>Randy, you are still too generous with compliments!
>
>Schlumberger is an oilfield service company. I do research and
>development
>for them. If I recall correctly, Schlumberger used to own Fairchild
>Semi in the
>late 70s, and sold them to National Semi.
>
>Jerry, I do not know about GE's connection to NBC, you probably know of
>a section or two that were bought/sold in the past.
>
>Schlumberger is an unusual choice for somebody with my background, but
>so far I have been very impressed with the company and I am having a
>lot of
>fun working on applied research for them! They own the entire "system"
>so
>I don't have to deal so much with standardization and they are very
>open to
>new ideas, no matter how unconventional it is.
>
>Cheers,
>Julius
Hi Julius,
Nice to hear from you. Hope ya' know
that we wish you all the best.
See Ya'
[-Rick-]
Reply by julius●December 11, 20062006-12-11
Randy Yates wrote:
> Good to see you here, Julius. Your presence boosts the groups
> IQ a couple of decades!
>
> Schlumberger - aren't they essentially what used to be National
> Semiconductor?
Randy, you are still too generous with compliments!
Schlumberger is an oilfield service company. I do research and
development
for them. If I recall correctly, Schlumberger used to own Fairchild
Semi in the
late 70s, and sold them to National Semi.
Jerry, I do not know about GE's connection to NBC, you probably know of
a section or two that were bought/sold in the past.
Schlumberger is an unusual choice for somebody with my background, but
so far I have been very impressed with the company and I am having a
lot of
fun working on applied research for them! They own the entire "system"
so
I don't have to deal so much with standardization and they are very
open to
new ideas, no matter how unconventional it is.
Cheers,
Julius
Reply by Jerry Avins●December 11, 20062006-12-11
Randy Yates wrote:
> "julius" <juliusk@gmail.com> writes:
>
>> Randy Yates wrote:
>>
>>> Julius,
>>>
>>> Did you see this which I posted a few days/weeks back? If so,
>>> why are you being so "anonymous"?
>>>
>>> Is this *THE* Julius Kusuma? Hey man, how's it going?! GREAT
>>> to see you back in comp.dsp. If you wanted to fill us in on
>>> your wanderings of late, we'd love to hear about it. Last I
>>> knew you were at MIT, but I'm thinking you've finished your
>>> PhD now.
>>>
>>> --Randy
>>>
>> Sorry, I didn't see the first time you posted this question. I did
>> finish
>> my PhD and I am now working for Schlumberger in Texas. I'll drop
>> in here every once in a while but since I now use google groups it's
>> too easy to miss postings, sorry!
>
> Good to see you here, Julius. Your presence boosts the groups
> IQ a couple of decades!
>
> Schlumberger - aren't they essentially what used to be National
> Semiconductor?
That;s like asking if GE is what used to be NBC. :-)
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Randy Yates●December 11, 20062006-12-11
"julius" <juliusk@gmail.com> writes:
> Randy Yates wrote:
>
>> Julius,
>>
>> Did you see this which I posted a few days/weeks back? If so,
>> why are you being so "anonymous"?
>>
>> Is this *THE* Julius Kusuma? Hey man, how's it going?! GREAT
>> to see you back in comp.dsp. If you wanted to fill us in on
>> your wanderings of late, we'd love to hear about it. Last I
>> knew you were at MIT, but I'm thinking you've finished your
>> PhD now.
>>
>> --Randy
>>
>
> Sorry, I didn't see the first time you posted this question. I did
> finish
> my PhD and I am now working for Schlumberger in Texas. I'll drop
> in here every once in a while but since I now use google groups it's
> too easy to miss postings, sorry!
Good to see you here, Julius. Your presence boosts the groups
IQ a couple of decades!
Schlumberger - aren't they essentially what used to be National
Semiconductor?
--
% Randy Yates % "Bird, on the wing,
%% Fuquay-Varina, NC % goes floating by
%%% 919-577-9882 % but there's a teardrop in his eye..."
%%%% <yates@ieee.org> % 'One Summer Dream', *Face The Music*, ELO
http://home.earthlink.net/~yatescr
Reply by julius●December 11, 20062006-12-11
Randy Yates wrote:
> Julius,
>
> Did you see this which I posted a few days/weeks back? If so,
> why are you being so "anonymous"?
>
> Is this *THE* Julius Kusuma? Hey man, how's it going?! GREAT
> to see you back in comp.dsp. If you wanted to fill us in on
> your wanderings of late, we'd love to hear about it. Last I
> knew you were at MIT, but I'm thinking you've finished your
> PhD now.
>
> --Randy
>
Sorry, I didn't see the first time you posted this question. I did
finish
my PhD and I am now working for Schlumberger in Texas. I'll drop
in here every once in a while but since I now use google groups it's
too easy to miss postings, sorry!
Julius
Reply by julius●December 11, 20062006-12-11
kl31n wrote:
> I couldn't find the discussion on the accuracy of the results depending on
> how many samples you use of the input signal. That's the part I'm
> interested in actually. As I said already maybe I'm not seeing the obvious,
> but would anybody be so kind to point me to a reference where not the
> problem is solved but it's solved with finite data sets and the accuracy of
> the results is discussed together with the general approach?
>
> Thank you again,
>
> kl31n
OK, I think I see where your hang-up is. I think the easiest way to
get
the answer is to write out your observation in terms of signal + noise,
for example:
y_n = A x_n + z_n. ... (1)
x_n will be your cos() and sin() in vector form, z_n is your noise, and
A
will be the vector of your wanted parameters.
In your original post you weren't clear whether the AWGN is in
continuous-time domain or in discrete-time domain. It will make a
big difference in how the variance of z_n will scale with T your
sampling
period. So you should clarify your model first.
Now, once you obtain (1) you can then apply your least-squares fit,
essentially by writing:
y = X a + z,
and obtaining the estimate \hat{a} from the pinv of X and the
observation y. This is called the Best Linear Unbiased Estimate, and I
would be very surprised if an estimation theory book does not cover
this
example.
Realize that X is completely known because you know what the frequency
is, so now you can also do a little experiment to see if the quality of
the
estimate of a depends on the frequency or not.
Hope this helps.
Julius
Reply by kl31n●December 5, 20062006-12-05
Il 4 Dec 2006 14:10:52 -0800, julius ha scritto:
> Vladimit's answer is very good, but note that you are comparing a
> parametric
> method (with a signal model) with a non-parametric method (only know
> that
> signal may be bandlimited). But since the wanted parameter in your
> signal
> appear linearly, then you are in luck and you can easily apply
> least-squares.
First of all thank you for you attention and your answer, however I'm not
trying to compare the two method, I'd simply like to understand which is
the accuracy with which I know the a1 and a2 terms once I solved the
overdetermined system using M samples of the input signal.
> If the parameter appears non-linearly, for example if you do not know
> the
> frequency, then it's a bit more complicated.
Such a problem is dramatically more complex.
> Like Vladimir said, the solution to the linear problem is a basic
> estimation
> theory problem, so any textbook would cover it.
I couldn't find the discussion on the accuracy of the results depending on
how many samples you use of the input signal. That's the part I'm
interested in actually. As I said already maybe I'm not seeing the obvious,
but would anybody be so kind to point me to a reference where not the
problem is solved but it's solved with finite data sets and the accuracy of
the results is discussed together with the general approach?
Thank you again,
kl31n
Reply by kl31n●December 5, 20062006-12-05
Il Sun, 3 Dec 2006 00:07:12 +0100, kl31n ha scritto:
> If I oversample a signal, I quantize it and then I reduce the sampling rate
> with a decimator by a factor M, I can improve the QSNR by a factor of
> log2(M).
Writing this I forgot to put (1/2) in front of the log2 funcion, so the
QSNR is actually improved by a factor
(1/2)*log2(M) = log2(sqrt(M)).
I guess the mistake was evident enough and that nobody fell for it, but to
not cause some useless confusion, I'll rather make the thread a bit more
chaotic and answer myself with such a correction.
kl31n