Thank you very much!

Fred Marshall wrote:
> (It's just been a long time since I read any of that stuff and I was never
> any good at it because matrix algebra was boring because the connection to
> the real world was almost always elusive.  I'm a slow learner in that way.
> I care about systems and not what may or may not be Hermitian, etc. etc. I
> imagine that Jerry might have a similar perspective).

Linear algebra *is* boring. I hate to admit it, but the reasons why I
got
interested, was that a fellow student of mine at college, who didn't
exactly
excel in maths, developed an interest in linear algebra and talked very

fondly about it. I got curious about what this guy had seen that I had
missed and took an extra intro class on linear algebra when I went to
university. That, along with matlab becoming available to me, basically

made it a no-brainer to start learning linear algebra.

After that, I used linear algebra in my MSc thesis basically to save on

writing efforts (a matrix equation is WAY easier to write than a
triple-
nested sum), as well as ease of programming (what can be written
as a matrix equation can be written almost directly as matlab code).

Heh, I started using linear algebra out of blows to my pride, and stuck

with it out of a highly developed sense of laziness...

Oh well.

Rune

erdinc.saygin@gmail.com wrote:
> Hi Rune,
>
> Well, I don't come from electrical engineering background so if I made
> a mistake in terms of jargon, sorry about that.
>
> It is an array of instrument which listen continously the earth.
>
> In this set up we have measurement between different points (assuming
> the noise is directional [from A to B]), then you can apply (try)  to
> estimate the response between these points.
>
> Of course, due to noise field , you may think this is a MIMO system but
> in that case, I don't know how to attack the problem. Any ideas will be
> appreciated.

I have seen similar problems being discussed in the books by
bendat and Piersol, and Bendat. They deal with SISO, MIMO, SIMO
etc systems, and show examples on how to analyze data.

I haven't really studied those sorts of things, since I have never had
an application for it, but it might be a starting point.

I'd suggest you find a copy of

Bendat & Piersol: Random Data, Wiley, 2000

and start  from there. 

Rune

Fred Marshall wrote:

> (It's just been a long time since I read any of that stuff and I was never 
> any good at it because matrix algebra was boring because the connection to 
> the real world was almost always elusive.  I'm a slow learner in that way. 
> I care about systems and not what may or may not be Hermitian, etc. etc. I 
> imagine that Jerry might have a similar perspective).

I plead guilty. :-)

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

<erdinc.saygin@gmail.com> wrote in message 
news:1143635865.670343.82260@i40g2000cwc.googlegroups.com...
> Hi Rune,
>
> Well, I don't come from electrical engineering background so if I made
> a mistake in terms of jargon, sorry about that.
>
> It is an array of instrument which listen continously the earth.
>
> In this set up we have measurement between different points (assuming
> the noise is directional [from A to B]), then you can apply (try)  to
> estimate the response between these points.
>
> Of course, due to noise field , you may think this is a MIMO system but
> in that case, I don't know how to attack the problem. Any ideas will be
> appreciated.
>

It seems to me, and I'm no expert in this, that you have a system with a 
bunch of boundary conditions "specified" i.e. measured.

If we have the boundary conditions can't we move away from the more usual 
idea of there being "forcing functions", "inputs" and "outputs"?  Isn't that 
what scattering coefficients in a network element are about to some extent? 
In that case what you observe at a port is a combination of the, shall I 
say, the *applied* input and what the system does to that input?

And the earth, in this case, is a "black box" (a thing with unknown insides) 
and let's assume that all the sources of energy are inside the box.  All we 
can do is observe the system at multiple "ports".  So, that's NIMO "no 
input, multiple output" because it's self contained with energy sources - 
not passive.

I'm not sure we have to commit to a lumped model of this distributed system. 
Anyway, here's a model:

Each "input" (it's unobservable and comes from inside the system) causes an 
output at each "port" or point of measurement.
There is a transfer function from each input to each of the outputs.
Accordingly, each output is a superposition of a multiplicity of inputs and 
transfer functions.
I can well imagine some system identification thesis dealing with this sort 
of thing and asking questions like how to separate out the inputs and 
transfer functions for each observation port.

(It's just been a long time since I read any of that stuff and I was never 
any good at it because matrix algebra was boring because the connection to 
the real world was almost always elusive.  I'm a slow learner in that way. 
I care about systems and not what may or may not be Hermitian, etc. etc. I 
imagine that Jerry might have a similar perspective).

Anyway, here we have this system of hidden sources and observation ports and 
their interconnecting transfer functions.
But, it's one system and not a network of filters if you will.  There is 
coupling between the observation ports that goes beyond interaction *at* the 
ports.  So the coupling has to be acccounted for and that suggests a 
distributed system model.

Perhaps I'm not being clear because at the top I refer to a model that's 
made up of blocks and here I'm referring to a model that represents the 
entire system (in whatever fashion works) - that is masses and springs and 
connecting rods and friction, etc.  It can be distributed or a finite 
element model or lumped which I guess is the same thing but on a gross scale 
perhaps.

A transfer function admits to a physical model.  So, reference to a transfer 
function alludes to such models.  So, when we talk about the model first, we 
are talking about a thing that will determine a transfer function or 
functions and coupling, etc.

Might the problem be approached by creating a model first?  Earth's crust 
thickness, density, etc. in a finite element grid.  Then complicate it with 
mountains and oceans as a second step to modify the distribution of mass, 
etc?  In fact, much of our analysis is based on assuming models first. 
Sometimes people aren't really aware of that and the model becomes their 
"truth".....

I'll bet it's reasonable to say:  "you can't have a notion of a transfer 
function without a model of some sort"

I'm not saying any of this because it's all that profound, simply that it 
may suggest a way forward.  I'll bet that somebody else here can jump in and 
help out.

Fred

Hi Rune,

Well, I don't come from electrical engineering background so if I made
a mistake in terms of jargon, sorry about that.

It is an array of instrument which listen continously the earth.

In this set up we have measurement between different points (assuming
the noise is directional [from A to B]), then you can apply (try)  to
estimate the response between these points.

Of course, due to noise field , you may think this is a MIMO system but
in that case, I don't know how to attack the problem. Any ideas will be
appreciated.

erdinc.saygin@gmail.com wrote:
> Hi all,
>
> I have a question about transfer function estimation relating
> averaging.
>
>
> The data that I have is daily records of measurement of earth's noise.
> The sample rate is 25 Hz. I have these records from multiple points and
> cross-correlating them to get the impulse response between these
> points(assuming that the noise has some "white" character).

"Multiple points"... multiple sensors in an array? Multiple arrays?

> With the
> original schema I have created some "respectable" results. But when i
> went into the details,I realized that it is not the best way to apply.
>
> Simply, I have segmented the each day with some overlapping and divided
> the corresponding part in frequency domain and then added them together
> in time domain.
>
> tft_final=ifft(tf(1))+ifft(tf(2))+.......ifft(tf(n)) where tf is the
> transfer function between each segments input (u) and output (u)
> (YU*/UU*) in frequency domain
>
> This is applied to all individual day records and then I averaged them.
>
>
> final_estimate=(1/n)(day1+day2+........dayn)
>
>
> But in terms of transfer function estimation, I don't think I have
> applied the right thing.

A "transfer function" relates an input to an output. Apparently, you
only
make measurements, you don't send. I can't really see how you can
measure a tranfer function from your particular set-up.

I suspect you are dealing with a Multiple-Input Multiple-Output system.

Multiple Output because you compare measurements from different
sensors. Multiple Input because the sensors may hear different noise
sources.

It is not obvious how to estimate a transfer function by comparing the
outputs of a MIMO system.

Rune

erdinc.saygin@gmail.com wrote:
...
>  If I summarize my question, "is averaging in frequency domain (Like
> Welch's Overlapped Segment Averaging) equal to averaging in time domain
> (after taking IFFT of each transfer function estimate)? Linearity says
> "yes" but not the results.

Welch's method specifies that the periodograms be averaged (the
magnitudes of the DFTed frames). This is not the same as time domain
averaging. If it were, it would by much simpler to do the time domain
averaging first, and apply only one DFT at the end. In short:

sum_k DFT[ x[n+k N] ] = DFT[ sum_k x[n+k N] ]
sum_k Abs( DFT[ x[n+k N] ] )  =/=  Abs( DFT[ sum_k x[n+k N] ] )

where N is the overlap length.

Furthermore, there are issues with the window used and the amount of
overlap from each successive frame. For some windows there exists an
overlap length such that time domain averging cancels out the window.
In general, however, windowed average of segments =/= average of
windowed segments.

Regards,
Andor

<erdinc.saygin@gmail.com> wrote in message 
news:1143507895.852230.105460@t31g2000cwb.googlegroups.com...
> Thanks for the response Fred. Well, it is seismic noise (no
> earthquakes). Without going into the details. Assume that it is an SISO
> system.
>
> If I summarize my question, "is averaging in frequency domain (Like
> Welch's Overlapped Segment Averaging) equal to averaging in time domain
> (after taking IFFT of each transfer function estimate)? Linearity says
> "yes" but not the results.
>
> Thank you very much in advance,
> Erdinc
>

Computing the average at each time and at each frequency over a bunch of 
sequences and not the average over time or the average over frequency seems 
OK.

On the other hand, note that the average of a time record over time is the 
frequency dc value or the value at zero frequency.
Note that the average of a frequency record over frequency is the the zero 
time value.
Now, these are very different.

Fred 

Thanks for the response Fred. Well, it is seismic noise (no
earthquakes). Without going into the details. Assume that it is an SISO
system.

 If I summarize my question, "is averaging in frequency domain (Like
Welch's Overlapped Segment Averaging) equal to averaging in time domain
(after taking IFFT of each transfer function estimate)? Linearity says
"yes" but not the results. 

Thank you very much in advance,
Erdinc