Fourier Transforms of chirp (swept sine wave) signals

[Jaglassc]

I am using a chirp signal as the excitation input to a piezoelectric actuator. I then measure the response (modified chirp signal). Both analogue signals are sampled via a DAQ board (PCI-MIO-XE-10). I am currently using the LabVIEW power spectrum VI to analyse the frequency domain of these signals. Given the characteristics of a chirp signal (low frequency at one end, high frequency at the other) there are major discontinuities when the waveform is repeated (as the Fourier transform assumes it does). Does this explain the oscillations seen on the frequency domain (Fourier) coefficients? Is there a better method of computing the FT of a chirp within LabVIEW? I have heard of a Fast Chirp Transform, does anyone know
any details of this algorithm and whether it can be/is implemented within LabVIEW?

[mikeporter]

A good solution for this type of analysis is the JTFA (Joint Time-Frequency Analysis) method developed at NI. It used to be a seperate toolkit, but now I'm not sure--check with NI.

Mike...

[LocalDSP]

A true time sweep will always show some ripple in the magnitude spectrum, especially near the ends of the spectrum (as you mention, low and high frequencies). The frequency domain ripple can be more or less attenuated using dedicated windowing and zero padding on your signal but this will also increase your signal length and therefore also your measurement time.

The question is: why are you using chirp signals in combination with FFT-based measurements? It sounds like you are mixing different technologies. A chirp is most often used when time continuous measurements are used (like rms as function of time with or without tracking filters), while FFT measurements are most convenient on time blocks that are perfectly periodic.

If you excitation signal
does not need to be a perfect time domain chirp, you can generate an approximated signal that "looks" almost like a chirp, but with a perfectly flat power spectrum (also when repeated). This signal will not result in any phase discontinuities. But nothing's free, so the price you are paying is:
- Not a perfect sweep (some cross-frequencies overlap)
- The crest factor is slightly higher than SQRT(2), that is the standard value for a sine tone or a swept sine tone.

The attached VI (LabVIEW 6.1 or higher) shows you how the high-level multi-sine generation VI can be used to create such a signal. If that type of signal can be used for the excitation of your actuator, then the Power Spectrum (or Frequency Response) VI will be perfectly suited for your application.

[urs boegli]

I guess jaglassc wants to measure the transfer function of the
'acoustic' line.
My hint would be to use a multitone like the one you mention. I would
prefer one with a different spacing of the frequencies, like one tone
per octave (f,2f,4f,8f...)
For the signalprocessing you could use methods like the lock-in
amplifier (special cross-correlation) and you will get a signal (x(t))
for each frequency content.
If you know the signal you have sent and measure what you get this will
be the much better approach than FFT and the like. By averaging over a
long time (seconds) you even can measure below the noise level
Good luck
Urs


LocalDSP schrieb:

>A true time sweep will always show some ripple in the magnitude
>spectrum, especially near the ends of the spectrum (as you mention,
>low and high frequencies). The frequency domain ripple can be more or
>less attenuated using dedicated windowing and zero padding on your
>signal but this will also increase your signal length and therefore
>also your measurement time.
>
>The question is: why are you using chirp signals in combination with
>FFT-based measurements? It sounds like you are mixing different
>technologies. A chirp is most often used when time continuous
>measurements are used (like rms as function of time with or without
>tracking filters), while FFT measurements are most convenient on time
>blocks that are perfectly periodic.
>
>If you excitation signal does not need to be a perfect time domain
>chirp, you can generate an approximated signal that "looks" almost
>like a chirp, but with a perfectly flat power spectrum (also when
>repeated). This signal will not result in any phase discontinuities.
>But nothing's free, so the price you are paying is:
>- Not a perfect sweep (some cross-frequencies overlap)
>- The crest factor is slightly higher than SQRT(2), that is the
>standard value for a sine tone or a swept sine tone.
>
>The attached VI (LabVIEW 6.1 or higher) shows you how the high-level
>multi-sine generation VI can be used to create such a signal. If that
>type of signal can be used for the excitation of your actuator, then
>the Power Spectrum (or Frequency Response) VI will be perfectly suited
>for your application.
>

[LocalDSP]

You are right, the transfer function can be measured with any type of multitone signal equally, exponentially or even randomly spaced (as long as you know your frequency list). In this particulary case, I was assuming that the "chirp" like excitation was required by the system. If it is not required, a dedicated multisine excitation and a true transfer function measurement (Frequency Response.vi) between the (re-)acquired excitation and response signals should work.
Furthermore if the generator can export a trigger signal or if the acquisition and processing can sustain real time, you may even consider to use vector (time signal) averaging to remove external uncorrelated noise.

[Jaglassc]

Thank you for your replies everyone. I am now using the multi-sine signal rather than the chirp which gives me a nice flat frequency response. I am however still observing ripple on the signal returned from the system (modified version of the multi-sine input signal). I am sampling at 5 times my highest frequency and the returned signal appears to be periodic (ie no discontinuities at the end points). What other effects could be producing this ripple? I do not understand how the Fourier Transform of the multi-sine produces a flat spectrum at all when I am using frequencies 1,2,3....,100 Hz. Most frequencies would not repeat periodically within the 1 sec time period - or is it only important that the multi-sine signal repeats perfectly?

[LocalDSP]

The ripple you see can be caused by different things.

You may, f.ex. have aliasing issues. I understand you are using a E10 board to generate and acquire you signals. On the generation side, the signal will include analog "images" of you original signal around the sample frequency and its multiples. These images will then be aliased back during acquisition and corrupt your result.

If you are measuring on an acoustic system, the ripple could also be caused by reflections, that is un-wanted addition of a delayed version of your signal. This will periodically be added in phase or in opposite phase, resulting in a smooth ripple (you can actually derive the reflection time from the ripple frequency).

So try to play with the s
ample rate (to see if it gets worse the lower you sample --> aliasing issues) or to modifiy your physical conditions (actuator position etc. --> external reflections or other problems)

Try to post your results so we can look at them (ripple size, ripple frequency, characterisitics etc..)