Network estimation with chirp signal.

[dora]

chirp signal and network analyzer

Hello All,

I am creating the simple network analyzer using the general purpose
DAQ board.

Currently I am testing the system using linear chirp (tone with sweeping frequency) excitation signal x. Then I correlate the output of the system y
with x and got estimation of the impulse system response h. FFT it to get
frequency response H

The procedure works because the autocorrelation of x is almost delta
pulse.

I am testing the procedure for several systems and it appears that levels
below -100 dB of the H are not estimated correctly. In other words I have
about 100 dB dynamic range.

I made compensation for the fact that autocorrelation of x is not ideal
delta pulse but still can not improve the accuracy.

I can use the white noise as stimulus and will get much below -100 db but
still think that the chirp should be better choice.

Opinions are well come.
Dora

 

[dora]

Hi Againg,

I put window on the input chirp signal and the dynamic range become more than 200 dB. The problem is that windowing the excitation chirp signal means reduction the lowest and higest frequency components. It means more noise in the network estimation at the low frequency and at the frequency close to the Nyquist rate.

Some ideas?
dora

 

[zorro]

Hi dora,

More than 200 dB !?
How is your system? And how do you measure this dynamic range?
Thanks

Z

 

[dora]

Hi Zorro,

Yes I was talking about simulation results in fact.
Since I am adapting my algorithm now.
I created Butterwort notch digital filter in Matlab and
test my algorithm for estimation of the frequency response.
I managed to get estimation deeply in the notch, -200dB below the DC level of the magnitude response.

You are right that it sounds very suspicious if it is the dynamic range of the real hardware. For the real analog filter frequency response even 70 db is too high to be archived.

regards
dora

 

[zorro]

Hi Dora,

OK. I guessed that it could be simulation results, but i got confused by the mention of a DAQ board.

I think that it is possible to expand the chirp bandwidth beyond the band of interest, in such a way that this whole band is far inough from the edges of the chirp.

There is another degree of freedom for the chirp control: the sweeping speed. It is possible to make a nonlinear frequency sweep, controlling in this way the autocorrelation function and the spectral contents. For instance, nonlinear frequency sweep (with a slower sweep at the start and the end) could compensate the decrease at the band edges due to windowing (amplitude shaping).
Nevertheless, thinking in Woodward’s theorem, I’m not sure that this can produce a big improvement. In any case, an increase in the product duration*bandwidth reduces the errors and increase the dynamic range.

Regards

Z

 

[dora]

#Hi Zorro,

OK. I guessed that it could be simulation results, but i got confused by the mention of a DAQ board.
#Yes I see

I think that it is possible to expand the chirp bandwidth beyond the band of interest, in such a way that this whole band is far inough from the edges of the chirp.
#Yes this is a solution but part of the full frequecy range will be lost.


There is another degree of freedom for the chirp control: the sweeping speed. It is possible to make a nonlinear frequency sweep, controlling in this way the autocorrelation function and the spectral contents. For instance, nonlinear frequency sweep (with a slower sweep at the start and the end) could compensate the decrease at the band edges due to windowing (amplitude shaping).
#Ahhh yes I see. I'll try this idea!

#In fact let me describe my current progress on this issue.
#First of all my experiments showed that the correlation method which I #wanted to use is not as better as simple division of the output to the #input spectrum. To have good dinamic range I use windowed chirp. The #problem with noise at the ends of the frequency grid appears.
#So I will try
#1. The approch with slower chirp at the ends. I'll try to generate the
# chirp with the modulating signal such that the magnitude responce is white. Probably modulating signal equal to the intergral of the window function will do the job.
#2. Frends of mine give me an idea to use triangle modulating signal
#(from 0Hz up to Nyquist and then down to 0 Hz ). This whay the #discontinuity probably can be removed and then most probably #windowing of the data will not be needed.

#Zorro what do you think about this ?

Nevertheless, thinking in Woodward’s theorem, I’m not sure that this can produce a big improvement. In any case, an increase in the product duration*bandwidth reduces the errors and increase the dynamic range.

#I didn't know about Woodward's theorem but found refference. Sounds logical indeed! Thanks Zorro for your competence!

#Best regards
#dora

 

[zorro]

Hi dora,

With triangular modulation you have two or more (say N) chirps in sequence, sweeping alternately with increasing and decreasing frequency. Observe that:

a) The product duration*bandwidth of them is 1/N of the value of a single chirp with the same duration and bandwidth.

b) Using a single chirp you have an amplitude discontinuity at the band edges, and this affects these frequencies. Using triangular modulation, you pass several times by each frequency, and just once or twice you have an amplitude discontinuity at a given edge. Nevertheless, there are always frequency discontinuities at the edges (phase continuity must be preserved) and I think that this can affect anyway the spectrum.

I'm not sure that triangular modulation gives an improvement. I think that a more accurate analysis can be performed. You can work with the autocorrelation functions of the chirps, that are mathematically treatable (if I’m not wrong, they involve Fresnel integrals).

Regards

Z

 

[dora]

Hello All,

I tested the approch with windowed chirp with instantenious frequency equal to the integral of this window.
This way the sweep speed is slower at the ends of the signal so more energy there.

The above idea produce good network estimation signal.
Unfortunately the magnitude responce of this signal is not flat.

So currently I am trying to descover the pair (instanteniours_freq(t), window(t)) which produces flat magnitude responce (other then the trivial non windowed linear chirp).

Any ideas?
dora

 

[panda]

What kind of network ure are gonna test?
Noone use that method with mobile channal, and
To test a filter, we use an diract impulse.

 

[dora]

Hi Panda,

I will use it for the telecommunication lines measurements.
I mean the copper lines with complex structure ... several connected lines with possibly bridge taps, etc.

But how do you use the Dirac delta , it is not realizable
what kind of approximation did you use?

thanks
dora

 

[zorro]

So currently I am trying to descover the pair (instanteniours_freq(t), window(t)) which produces flat magnitude responce (other then the trivial non windowed linear chirp).

Any ideas?
dora

Take into account that it is not possible that a signal be both time limited and have a rigorously flat amplitude spectrum in a frequency band. A certain degree of distortion must be tolerated.

In other words: you want a signal with flat amplitude spectrum. Such a signal has an autocorrelation function that extends from -Inf to + Inf. But this autocorrelation function cannot correspond to a finite duration signal.

Regards

Z

 

[dora]

Hi zorro,

In fact I meant that it should be flat in terms of frequency grid used by the FFT (as the classical linear chirp is). In fact the FFT produces the decomposition in terms of periodic functions with main period equal to the observation interval. So it does extend my chirp from -inf till +inf.
In fact not perfectly flat spectrum is OK also.

The next problem is howevar that I have feeling that the chirp is not the best exitation signal. I perform some preliminary test with standard sound card (still DAQ card is not in house) and it seems for me that chirp signal makes the procedure very sensitive to phase noise (shifting of the input and output signal vectors).

I have to test more to say something more concrete.

The problem is that I don't have good book about network estimation.
And now I am tring to descover the wheel.

Best regards
dora