[penguin121]

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Originally Posted by JaZZ /img/forum/go_quote.gif I agree. But «good» frequency response is not perfect frequency response. I was talking of the latter – from a merely theoretical perspective. However, a fairly flat frequency response will guarantee a fairly good phase and transient response. (Still with respect to full-range transducers.) .

I am still not getting your basis for this statement. Keep in mind that the frequency spectra is flat for both ideal white noise and an ideal impulse (delta function). The entire difference between these two signals is contained in the relative phase (linear for the impulse, random for the white noise). If you were only to look at the amplitude of the frequency response you would not be able to distinguish the two. I don't doubt that drivers designed to give flat frequency response are most likely also designed to provide accurate phase. However, I see no reason to believe that flat frequency response necessarily guarantees accurate phase response, and therefore good transient response.

 

[JaZZ]

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Originally Posted by b0dhi /img/forum/go_quote.gif I'm curious as to why you believe this?

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Originally Posted by penguin121 /img/forum/go_quote.gif I am still not getting your basis for this statement. Keep in mind that the frequency spectra is flat for both ideal white noise and an ideal impulse (delta function). The entire difference between these two signals is contained in the relative phase (linear for the impulse, random for the white noise). If you were only to look at the amplitude of the frequency response you would not be able to distinguish the two. I don't doubt that drivers designed to give flat frequency response are most likely also designed to provide accurate phase. However, I see no reason to believe that flat frequency response necessarily guarantees accurate phase response, and therefore good transient response.

Frequency spectra (measurements of fourier components) are not the same as frequency response. For measuring phase response you need a periodic signal, and for measuring frequency response you need sine waves or a sine sweep. – Note that we're not interested in the frequency content of the original signal (noise, thus an erratic signal, in your case), but we want to examine the transfer function of a component. This for we need clean, reproduceable signals.

During my speaker-builder career I've dealt with crossover networks. There are formulae for different types of low- and high-pass filters: from 1st- to 4th-order filters with 6 to 24 dB/oct. drop-off. Every filter type generates a specific phase distortion at the -6-dB point (the crossover frequency): 90° for 1st-, 180° for 2nd-, 270° for 3rd-, 360° for 4th-order filters. You can't have a low- or high-pass filter without this accompanying phase distortion.

The same applies to equalizers. Set to neutral for a flat frequency response, they also have a flat phase response (given unlimited bandwidth). Every distortion of the frequency response comes with a corresponding distortion of the phase response. You can even observe this phenomenon with a sound editor: emphasizing the bass with the equalizer leads to a migration of the bass waves towards the right, reducing the bass leads to a migration to the left.
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[penguin121]

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Originally Posted by JaZZ /img/forum/go_quote.gif Frequency spectra (measurements of fourier components) are not the same as frequency response. For measuring phase response you need a periodic signal, and for measuring frequency response you need sine waves or a sine sweep. – Note that we're not interested in the frequency content of the original signal (noise, thus an erratic signal, in your case), but we want to examine the transfer function of a component. This for we need clean, reproduceable signals..

While frequency spectra is not the same as frequency response, the point was that frequency response amplitude is not sufficient to describe the performance of transducer as is does not account for all of the factors the affect the signal reproduced. Also, I assume that by frequency response, you mean frequency response amplitude, as frequency response is in fact a complex quantity with both amplitude and phase.

As for measuring the transfer function of a transducer, there are a number of different ways to do this in addition to those you already mentioned. These include methods that allow for both the amplitude and phase of the frequency response to be measured simultaneously. These work by recording time-synchronous measurements of both the input and the transducer response, allowing for the complex transfer function to be directly calculated. The input would typically be bandpass filtered white noise, though other broadband input signals could also be used if desired.

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Originally Posted by JaZZ /img/forum/go_quote.gif During my speaker-builder career I've dealt with crossover networks. There are formulae for different types of low- and high-pass filters: from 1st- to 4th-order filters with 6 to 24 dB/oct. drop-off. Every filter type generates a specific phase distortion at the -6-dB point (the crossover frequency): 90° for 1st-, 180° for 2nd-, 270° for 3rd-, 360° for 4th-order filters. You can't have a low- or high-pass filter without this accompanying phase distortion. The same applies to equalizers. Set to neutral for a flat frequency response, they also have a flat phase response (given unlimited bandwidth). Every distortion of the frequency response comes with a corresponding distortion of the phase response. You can even observe this phenomenon with a sound editor: emphasizing the bass with the equalizer leads to a migration of the bass waves towards the right, reducing the bass leads to a migration to the left. .

So are you saying that because applying a well designed crossover filter produces a certain phase distortion, that anything exhibiting the same frequency response amplitude as that filter produces will have the same phase distortion? I don't think that this is valid to assumption to make, as those filters are specifically designed to have those characteristics. The phase distortions being what they are as a result of the trade-offs needed to balance the filter design for the specific application at hand. For any well designed filter that produces a certain frequency response, there are many ways to create a poorly designed filter that will produce a frequency response that maintains the same amplitude as the well-designed filter, while completely messing up the phase of the frequency response. I still see no reason to expect that the relationship between the phase and the amplitude of the frequency response for a transducer should follow that of a crossover or EQ filter.

 

[JaZZ]

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Originally Posted by penguin121 /img/forum/go_quote.gif While frequency spectra is not the same as frequency response, the point was that frequency response amplitude is not sufficient to describe the performance of transducer as is does not account for all of the factors the affect the signal reproduced.

I'm not completely sure about that, but I think you can perfectly calculate phase and transient behavior out of a frequency response. However, measuring resolution may be a limiting factor. The opposite is true by all means.


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Also, I assume that by frequency response, you mean frequency response amplitude, as frequency response is in fact a complex quantity with both amplitude and phase.

Sure do I mean amplitude response.


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As for measuring the transfer function of a transducer, there are a number of different ways to do this in addition to those you already mentioned. These include methods that allow for both the amplitude and phase of the frequency response to be measured simultaneously. These work by recording time-synchronous measurements of both the input and the transducer response, allowing for the complex transfer function to be directly calculated. The input would typically be bandpass filtered white noise, though other broadband input signals could also be used if desired.

Yes, there are several ways of measuring frequency, transient and phase response. As mentioned, you can calculate the whole from a single step or pulse response (and get a waterfall plot plus frequency and phase response that way). I was talking of sine waves in the context of your white-noise example. Because you can't measure the phase response with band-pass filtered white noise, you can't even measure the exact frequency response that way – as the noise still contains a certain frequency band, so the measurement is smeared, which leads to a smoothed frequency response (still suitable for speakers and headphones).


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So are you saying that because applying a well designed crossover filter produces a certain phase distortion, that anything exhibiting the same frequency response amplitude as that filter produces will have the same phase distortion?

Yes, that's what I'm saying. The crucial point is that you can't have a distorted frequency response without a (accordingly) distorted phase response: the phase distortion will always correspond 100% to a specific frequency-response distortion. That's why a «poorly designed filter» (which may be the ideal filter for a specific application nonetheless) with a deviating phase response necessarily has a deviating filter slope. You could read the corresponding literature if you want in-depth information.


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I still see no reason to expect that the relationship between the phase and the amplitude of the frequency response for a transducer should follow that of a crossover or EQ filter.

Instead of just disbelieving you could do the proposed sound-editor experiment!

It's important to know that you can't have frequency-response distortion without phase-response distortion. A crossover filter is just a normal filter, nothing else. It is a good example because of the standardized formulae for a specific filter characteristic for both frequency and phase response. Now you could design a -6-dB drop-off at the reference frequency with a different phase distortion than standard; but this would also mean a deviating frequency response (and vice versa). There are many cases where it's appropriate to modify the standard characteristics, because you also have to take the transducers' transfer function into consideration – and both add together.

Fascinatingly you can compensate electronically for many transducer shortcomings. Theoretically you can completely eliminate even severe resonances – by mere band-pass equalizing. It not only elminates the amplitude excess, but also the delay of decay. All you need is an inverse transfer function. Or you can compensate for the bass drop-off of a closed speaker. At the same time the transient response gets improved and the bass resonance eliminated (for a perfect result you need a perfect correlation, though).

As to the equation of electronics and sound-transducer transfer function, search for «signal theory».
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[penguin121]

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Originally Posted by JaZZ /img/forum/go_quote.gif I'm not completely sure about that, but I think you can perfectly calculate phase and transient behavior out of a frequency response. However, measuring resolution may be a limiting factor. The opposite is true by all means.

And it turns out you would be correct! I was thinking of this in a more general sense, and generally speaking phase is also required to calculate transient response as the frequency response amplitude only tells you the relative magnitude of the Fourier components, but not how those components line up in time. However, for a minimum-phase system the phase response can be exactly calculated from the amplitude response by means of a Hilbert transform. So, as it turns out, the detail that I was missing is that a speaker driver is in fact a minimum-phase system!

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Yes, there are several ways of measuring frequency, transient and phase response. As mentioned, you can calculate the whole from a single step or pulse response (and get a waterfall plot plus frequency and phase response that way). I was talking of sine waves in the context of your white-noise example. Because you can't measure the phase response with band-pass filtered white noise, you can't even measure the exact frequency response that way – as the noise still contains a certain frequency band, so the measurement is smeared, which leads to a smoothed frequency response (still suitable for speakers and headphones).

I have to disagree here. You can measure both the phase and amplitude of the frequency response over the passband using band-pass filtered white noise, provided the correct approach is used. It sounds like you are describing the result that you would get by using the square root of the ratio of the RMS-averaged auto-spectra of the two channels to estimate the transfer function. A better way to estimate the transfer function in this case would be to use the ratio of the averaged cross spectrum to the averaged auto-spectrum of the input signal. The cross-spectrum has the advantage of maintaining the relative phase between the two channels, which should remain constant. In addition to allowing for the complex transfer function to be approximated, this approach also offers better noise rejection in this approximation, which will converge to the exact transfer function if enough records are averaged.

 

[JaZZ]

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Originally Posted by penguin121 /img/forum/go_quote.gif And it turns out you would be correct! I was thinking of this in a more general sense, and generally speaking phase is also required to calculate transient response as the frequency response amplitude only tells you the relative magnitude of the Fourier components, but not how those components line up in time. However, for a minimum-phase system the phase response can be exactly calculated from the amplitude response by means of a Hilbert transform. So, as it turns out, the detail that I was missing is that a speaker driver is in fact a minimum-phase system!

Exactly: minimum-phase system is the keyword!

(Almost forgot the term.)


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I have to disagree here. You can measure both the phase and amplitude of the frequency response over the passband using band-pass filtered white noise, provided the correct approach is used. It sounds like you are describing the result that you would get by using the square root of the ratio of the RMS-averaged auto-spectra of the two channels to estimate the transfer function. A better way to estimate the transfer function in this case would be to use the ratio of the averaged cross spectrum to the averaged auto-spectrum of the input signal. The cross-spectrum has the advantage of maintaining the relative phase between the two channels, which should remain constant. In addition to allowing for the complex transfer function to be approximated, this approach also offers better noise rejection in this approximation, which will converge to the exact transfer function if enough records are averaged.

I think I halfways get what you're saying (not sure, though). However, measuring techniques are a side scene which doesn't interest me that much, I'm more fascinated by a holistic view of audio phenomena. An «audio TOE», so to speak.

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[b0dhi]

Hehe yeah, speakers are minimum phase, just as they are pistonic point sources