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What is Power Cepstrum and how is it useful for speaker measurements?

nahuel

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Hi, I've seen this graph of amplitude vs quefrency and, after reading wikipedia, still couldn't figure it out, so I thought I would ask for help here.

How is it useful for speaker measurement? Other uses should be interesting to know.

Literature recommendations for starters are very welcome!

Thanks! :)
 

dc655321

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I've never used cepstrum as an analysis tool, but I have used the same math to reconstruct the phase spectrum given an amplitude spectrum and an assumption of a minimum phase system.
 

pkane

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Here's an example.

Echo of 0.1s was added using the Echo effect in Audacity to a music track. Track then processed using DeltaWave to display the Cepstrum:
1631815373333.png


You can see that there was the main echo at 0.1s, and then a much reduced echo at 0.2s, followed by a tiny one at 0.3s. You can almost see the one at 0.4s.
 

Langston Holland

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I've seen this graph of amplitude vs quefrency and, after reading wikipedia, still couldn't figure it out,

You and me both. : ) The math eludes me, but I think I can help from an application point of view, though Paul answered perfectly above.

I use the Cepstrum plot along with the IR and to a lesser extent, the ETC in loudspeaker design. I find it indispensable like a weird type of screwdriver that fits a weird type of screw I run into frequently. Ever work with Neutrik NL4 connectors? Find that your #1 Phillips screwdriver slips out when finalizing the torque on the cable? That's because you need a #1 "Pozidriv" screwdriver instead. It'll change your life.

I advise you to stop reading now and just remember what Paul said because I'm going to get into it. : )

---

Cepstrum, like the periodic impulse response (PIR or IR), like log squared, like the energy time curve (ETC) are all ways of looking at the arrival of sound over time. Each have advantages for given situations, which is why it's nice to have access to each when interpreting the time domain behavior of sound.

I could be evil and say something like "the Cepstrum is the inverse FFT of the Log Squared representation of the IR with phase preserved", but I would never stoop to that! The Cepstrum preserves phase and - importantly - removes noise from the measurement for a much more detailed view of arrival events. Once you have a handle on the IR and Log (most people leave off the word squared) responses, I think you'll find the Cepstrum response is pretty intuitive - and you'll wonder why the heck it is so hard to find in measurement software! Don't even get me started about the Nyquist plot..

Sound is the variation of air pressure above and below atmospheric pressure.

The IR is a representation of the measurement microphone's diaphragm movement in response to the sound pressure variation it was exposed to. If all goes well, the trace above zero is the positive pressure when the mic diaphragm was pushed inward and the below zero trace is when the diaphragm is pulled outward. Thus phase is preserved in its display. The IR is a mess with acoustics work and indispensable with loudspeaker design.

The Log takes the IR data, squares it (to get rid of negatives 'cause logs don't like negative numbers), then presents this data normalized to 1, meaning that all the wiggles are relative to the peak arrival instead of an absolute number of pascals or SPL (the ETC is usually normalized to 1 as well). The Log response will always be positive. It is helpful with acoustics and loudspeaker design, but the IR, ETC and Cepstrum are better, thus obviate the need for it.

Heyser's ETC is kind of like the Log response, but has some really fancy stuff going on to more accurately display the total energy of the sound arrivals over time. Phase is not preserved (technically it is, but it isn't shown without a 3D plot with a Z axis). It's indispensable in acoustics work and helpful with loudspeaker design.

This is a comparison from the Smaart v7/v8 manuals of the IR, Log and ETC
Smaart.png


Here's the frequency domain mess, I mean response of a 1986 vintage Klipsch Heresy II loudspeaker
Heresy.png


This is the IR time domain response of this measurement
IR.png


This is the ETC time domain response of this measurement
ETC.png


This is the Cepstrum time domain response of this measurement
Cepstrum.png


God bless you and your precious family - Langston

Extra Credit: So where did the term Cepstrum come from? Cepstrum obviously reverses the first (4) letters of Spectrum. This is because Cepstrum is the inverse spectrum of the Log spectrum. Due to the inversion, the Cepstrum frequency domain was coined Quefrency and harmonics were coined Rahmonics.
 
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dc655321

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I could be evil and say something like "the Cepstrum is the inverse FFT of the Log Squared representation of the IR with phase preserved", but I would never stoop to that! The Cepstrum preserves phase

In what sense is "phase preserved"?
The Real Cepstrum, as you're illustrating, uses only the magnitude (hence discards phase spectra).
 

Langston Holland

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The Real Cepstrum, as you're illustrating, uses only the magnitude (hence discards phase spectra).

Incorrect. Time domain Cepstrum such as shown above necessarily is complex. Complex means magnitude and phase. Thus phase is preserved.

Randall with Bruel & Kjaer introduced the audio world to complex Cepstrum analysis in 1981. He did a review of the progress made with further applications in 2017. Both links are provided below. From the latter:

It was a small further step to perform a (linear) inverse Fourier transform on the log spectrum to obtain a new type of cepstrum, and this was the basis of Schafer’s PhD dissertation at MIT [6]. By retaining the phase in all operations, the “complex cepstrum” was defined as the inverse Fourier transform of the complex logarithm of the complex spectrum, and this was thus reversible to the time domain.

From the designers of the software I used to present the time domain Cepstrum:

A not so common calculation, but very useful is the Cepstrum analysis. The benefits over PIR view is that the noise is suppressed, so reflections are clearly visible [19, 20]. This is very useful for setting the time windows. Cepstrum is the inverse FFT of the logarithmic magnitude with the phase preserved. As the logarithmic scale doesn’t contain negative values, the time domain will be “rectified”. The peak is normalized to one.

Links
1. 1981 B&K Publication
2. 2017 Univ. of NSW Publication
3. 2010 WaveCapture Bävholm & Grenander ”Delay Alignment; A Survey”

Maybe you could contact the principle designer of the software (Johny Grenander) and inform him of his error.
 
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nahuel

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Hey, you are all great, thank you so much. I need to read all this again tomorrow, but i'm starting to get a sense of it.

You're the best and this is good fun stuff :) I guess i'll ask again tomorrow though. I see that coming :/

Blessings to you all! Beautiful geeky people, i'll be like you someday.
Nahuel
 

dc655321

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Incorrect. Time domain Cepstrum such as shown above necessarily is complex. Complex means magnitude and phase. Thus phase is preserved.

I see. You're almost right - the Complex Cepstrum is constructed with the phase of the original signal, but produces a Real result.
I'm afraid I don't know what a Time domain Cepstrum is...

Maybe you could contact the principle designer of the software (Johny Grenander) and inform him of his error.

God bless your precious facetiousness. :rolleyes:
 
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