Transfer Function Estimation - Averaging?

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). 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. Then I used the same data and tried windowing
and averaging in frequency domain with same amount of segmenting and
overlapping. Then came back to time domain. But the results are simply
too noisy and lacking some information (e.g acausal part of the signal
didn't carry the expected information unlike the first case).

tft_final=ifft(tf(1)+tf(2))+.......tf(n))/n

then
final_estimate=(1/n)(day1+day2+........dayn)


My question is, what might have I have done wrong. By the property of
linearity of DFT, I was expecting the same results for the both cases.


Thanks in advance,

Erdinc

<erdinc.saygin@gmail.com> wrote in message 
news:1143468591.878807.199380@j33g2000cwa.googlegroups.com...
> 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). 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. Then I used the same data and tried windowing
> and averaging in frequency domain with same amount of segmenting and
> overlapping. Then came back to time domain. But the results are simply
> too noisy and lacking some information (e.g acausal part of the signal
> didn't carry the expected information unlike the first case).
>
> tft_final=ifft(tf(1)+tf(2))+.......tf(n))/n
>
> then
> final_estimate=(1/n)(day1+day2+........dayn)
>
>
> My question is, what might have I have done wrong. By the property of
> linearity of DFT, I was expecting the same results for the both cases.

Google on "system identification" to find discussion and methods.

I'm not sure what you're doing.  Is this seismic noise?
Assuming that's the case then what are you using for an *input*?
It seems to me that you have a multi-input, multi-output system (really a 
distributed system which might more often be represented by partial 
differential equations.  But, OK, you've got a lumped model and are using 
ordinary differential equaations it appears.
Anyway, it looks like you're working with nothing but "outputs" so that's 
curious.
How do you separate things out?

I guess this might be a bit like a lumped network model where inputs are 
also outputs and outputs are also inputs?  I've touched on those things in 
school but never used them in practice.  How do you separate out the "input" 
part of a measurement (e.g. local seismic activity) from the "output" part 
(response from distant activity)?

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

<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 

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:
> 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

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 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

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:
> 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