Does Phase Distortion/Shift Matter in Audio? (no*)

[tengiz]

Define two pink noise signals, A and B, each independent of the other.

...

I've just tried this:

Let left = a + cos (x) * b
Let right = a + sin (x) * b

where x = n / 88100, and n is the sample number assuming 44.1kHz sampling rate.

Going back to the absolutely useless topic (in the context of speaker design), can we take the Time domain impulse of the 1 sample at 48kHz and run an FFT on that?

Or push that through a DAC that band limits it below Nyquist…
And then run that into a ADC that samples at 96, 192 ?
… Would that then be 2 or 4 samples wide?

A single non-zero sample file ignores the headroom for practical anti-aliasing/reconstruction filter roll-off and runs full-band right up to Nyquist. Same, of course, with Amir’s random-sample file. Both are "illegal" in this sense.

So it's highly unlikely to see a single non-zero sample file (except if the usable bit depth is so low that quantization forces it).

...Unless, of course, you specifically doctor the spectrum of the input pulse in the analog domain to compensate for your ADC’s anti-aliasing filter . In theory, you could manipulate the highs so it comes out the other end with exactly one non-zero sample and if you somehow also are able to time it with the precision that it would require - just don’t pump too much near Nyquist or the front end might have a fit .

 

[pma]

A single non-zero sample file ignores the headroom for practical anti-aliasing/reconstruction filter roll-off and runs full-band right up to Nyquist. Same, of course, with Amir’s random-sample file. Both are "illegal" in this sense.

A single non-zero sample is a mathematical construct in the digital samples domain, it is perfectly valid and has uniform spectrum from DC to Fs/2. Below such signal with Fs = 96kHz. It goes to exactly 48kHz. Do not mess with the reconstruction filter here. Such filter is an additional implementation to get time domain signal from the samples.

 

[tengiz]

A single non-zero sample is a mathematical construct in the digital samples domain, it is perfectly valid and has uniform spectrum from DC to Fs/2. Below such signal with Fs = 96kHz. It goes to exactly 48kHz. Do not mess with the reconstruction filter here. Such filter is an additional implementation to get time domain signal from the samples.

How did you obtain the 1_impulse_96kHz_16bit.wav? Did you synthesize it or did you digitize an analog source of the pulse? It's not "illegal" to have anything in a digital file, but the practical transition between digital and analog domains imposes limitations.

 

[j_j]

I've just tried this:

Let left = a + cos (x) * b
Let right = a + sin (x) * b

where x = n / 88100, and n is the sample number assuming 44.1kHz sampling rate.

I don't mean to modulate or keep constant energy, I mean to add increasing amounts of antiphase to the inphase, while listening binaurally. But this certainly has effects, too!

 

[j_j]

...Unless, of course, you specifically doctor the spectrum of the input pulse in the analog domain to compensate for your ADC’s anti-aliasing filter .

If you do that, then you have energy at fs/2

So you haven't done proper antialiasing after all. Ooops.

 

[amirm]

A single non-zero sample is a mathematical construct in the digital samples domain, it is perfectly valid and has uniform spectrum from DC to Fs/2.

So it is your opinion that the identical waveform in time domain, has infinite frequency domain representations depending on value of sampling rate? If so, the universe would not exist as it does today!

Let's say your sample rate is 20/sec. Half of that would be 10 Hz. Is it your opinion that an amplifier can reproduce such a spike with just 10 Hz of bandwidth?

Below such signal with Fs = 96kHz. It goes to exactly 48kHz. Do not mess with the reconstruction filter here. Such filter is an additional implementation to get time domain signal from the samples.

No. You are again confused with what an audio workstation software shows vs reality. That truncation at half the sample rate is artificial feature of the software. Physics of that impulse doesn't care what number you put in the file as its sample rate. You cannot reproduce that signal without infinite bandwidth. Or else, rise time would not be proportional to bandwidth which it is :

If RT is zero, you get a divide by zero and Bandwidth would be infinite.

Really, we have explained this before. There is a reason Nyquist theorem exists. What you create must be bandlimited and an impulse is not.

 

[tengiz]

If you do that, then you have energy at fs/2

So you haven't done proper antialiasing after all. Ooops.

Exactly - and that’s what makes a digital file with no consideration for it "illegal". They exist, but with what a physicist would call a "spherical cow in a vacuum" kind of twist.

 

[pma]

Really, we have explained this before. There is a reason Nyquist theorem exists. What you create must be bandlimited and an impulse is not.

We are speaking about digital to analog, not about ADC with anti-aliasing input filter. It is bandlimited by a DAC reconstruction filters, it is bandlimited by Adobe Audition brickwall display filter for viewing. How else would you check DAC impulse response than with a single impulse digital file?? How would you check DAC step response than with a similarly created unit step file?? If you test with Fs/4 (sine) signal, the digital file used has exactly 4 non-zero samples per period, with same amplitude, all other samples being zero. The transition between sample values is again sudden, same as in case of a single non zero sample. We are in a digital sample domain. Do not confuse it with time or frequency domain. How is it possible that if you test DAC with white noise or a sine close to Fs/2 you almost always see attenuated mirror images above Fs/2? Would you call it illegal, again? No, it is a filter leakage above Fs/2. Absolutely same case.

 

[amirm]

We are speaking about digital to analog, not about ADC with anti-aliasing input filter. It is bandlimited by a DAC reconstruction filters, it is bandlimited by Adobe Audition brickwall display filter for viewing.

That's right. You post an output that had an artificial, software created view as representation of reality. That was quite wrong.

How else would you check DAC impulse response than with a single impulse digital file?? How would you check DAC step response than with a similarly created unit step file??

You wouldn't. The impulse response is useful if you are trying to design/analyze a digital filter. Using that representation, you can build say, a low pass filter in time domain.

No one trying to assess the fidelity of audio gear should resort to such tests unless they are specifically examining filter response and such. Otherwise, they will run all concerned thinking every transient in their music looks like a SINC function:

The first row shows that a band limited system will have that Sinc() response, should it be excited with full force of its spectrum. In real music, we are not at all facing that so such a response/distortion does not exist.

The second row shows what happens if you design a filter using too few coefficients. If you were interested in such things, then sure you can use an impulse response.

Showing those time domain responses is not going to be useful to anyone wanting to go how good a DAC is. You can't, without performing the actual math, derive the frequency response. This is why I provide frequency domain analysis using noise as the source signal:

If I showed you the time domain responses of those filters, you would have no prayer of getting the above understanding.

Bottom line: if you know signal processing and want to create digital filters and such, understanding impulse response is mandatory. For audio review/fidelity analysis, it has little to no use.

 

[pma]

This has become annoying. I have never denied validity of Nyquist sampling theorem, but that theorem says that the signal with BW < Fs/2 can be completely reconstructed from its samples taken at Fs. It is a sampling theorem and speaks about proper sampling of the input signal.

Which is apples and oranges. I have always been speaking about mathematically created signal in a digital domain and about its (proper) conversion to analog signal, not about sampling of the input signal. The signal mathematically created and addicted to Fs sampling frequency can never contain frequency components above Fs/2, even if it is a single non-zero impulse. It is just impossible to have >Fs/2 frequencies and the signal violates nothing. If the DAC analog output is spread above Fs/2, then it is a result of non-perfect implementation only, namely the output brickwall filter. If you happened to read the DSP book:

you might like read definitions first. And they define impulse and impulse response (of a digital system as well)

So, for discrete signal, impulse is all zeros and a single non-zero impulse. It is not any "illegal signal" as you are trying to persuade your non-qualified audience. And impulse response is a response to this normalized impulse - delta function. Should it be a single sample or continuous delta function (this is a technical term again). Impulse response of a digital D/A system system cannot be tested by some broadened strange impulse signal. So, your position is indefensible.

 

[j_j]

The point, simply put, is that you can never generate that with a valid "INPUT" to the INPUT anti-aliasing filter. You can certainly write a program that goes x(1:100000,1:2)=0;
x(3,1:2)=.9999 followed by audiowrite('tick.wav',48000,'BitsPerSample', 24)

BUT you can not generate that file from the input antialiasing filter.

Nobody is arguing that an impulse is not an impulse, the question is "how do you get one at this point in the system, USING THE SYSTEM". Yes, it does ok for measuring the system (but there are better choices!), and you also want to keep that signal down at least 8dB just to avoid various "things" in an anti-imaging filter that might not expect that power at pi/2.

 

[pma]

BUT you can not generate that file from the input antialiasing filter.

This is something I have never argued with, and I fully agree, but it is about ADC. However, I WANT to test DACs with a discrete delta function and with a discrete step:



and with other "silly" discrete signals, like this one (again contains sample-to-sample almost FS changes):

These wav signals were burnt to CD and ripped from CD. Then they are compared to DAC or CD player analog output., sampled at much higher frequency than 44.1kHz.

 

[j_j]

This is something I have never argued with, and I fully agree, but it is about ADC. However, I WANT to test DACs with a discrete delta function and with a discrete step:

View attachment 470048

and with other "silly" discrete signals, like this one (again contains sample-to-sample almost FS changes):

View attachment 470061

These wav signals were burnt to CD and ripped from CD. Then they are compared to DAC or CD player analog output., sampled at much higher frequency than 44.1kHz.

I understand your testing method, that's much like the "buzz tones" and various tests intended to see what happens with intersample overs, which are still a problem in a lot of DAC's. A lot of equipment twitches violently with highly asymmetric content.

 

[amirm]

The signal mathematically created and addicted to Fs sampling frequency can never contain frequency components above Fs/2, even if it is a single non-zero impulse.

Of course it can. I show this in many DAC reviews with computer generated noise signal with much higher bandwidth than half the sampling rate:

Notice the response in blue which keeps going and going. I specifically use this test signal because it can exceed half the sampling the rate. AP's software generator will refuse to create such illegal signal because it respects what can properly be digitized. Using noise signal gets around that.

Above also shows why there is limited use for such signals. I explained how in filter design you want to use impulse response as it has infinite bandwidth so can properly characterize any filter (in that domain). That is why your signal processing book talks about it. As they say though, there is a difference between what is in a textbook and what is reality. It is the latter that I and JJ are trying to explain to you.

Nobody here is using a DAC by itself, absence of generation by its inverse, i.e. ADC if they are listening to music. If the signal cannot be naturally produced, then you are creating specific test cases to serve another need, like I am doing with the above noise signal. Be clear about it and there is no issue. Run off and demand that products do this and that in the face of illegal input and we will have word.

Bottom line, you absolutely can create out of band files with computer manipulation. It is easily demonstrated as I show above, and understood by knowing Fourier transform of such signals. Such pathological tests can have uses, but you best know what they are and not generalize.

 

[amirm]

Impulse response of a digital D/A system system cannot be tested by some broadened strange impulse signal. So, your position is indefensible.

Sorry, no. Our application for music is analog to analog. No one listens to digital data. We represent analog music in digital domain for ease of transmission but the two end points are analog. To do that, you must have two components per Nyquist theorem:

1. Input signal must be band limited to half the sample rate.

2. Output signal must be band limited to half the sample rate.

If you do these two perfectly, then the digital system is transparent, end to end.

You have skipped over #1, created an artificial signal that by definition, has infinite bandwidth. Here is the Fourier transform of an impulse/delta function:

Notice the text, "All frequencies applied to filter input with equal intensity." Again, "all frequencies." That is the core thesis behind an impulse function. That is, it excites the system at all frequencies. Therefore, what we see represents all possible frequencies and hence, is the entire response of *any* system regardless of its bandwidth.

It is therefore simple to see that the impulse signal violates Nyquist theorem and therefore is an illegal signal. Is it useful for some things? Absolutely. You just need to understand what it is, and where you are going outside the bounds of system fidelity.

 

[bmc0]

Of course it can. I show this in many DAC reviews with computer generated noise signal with much higher bandwidth than half the sampling rate:

Perhaps I'm missing something, but what you seem to be claiming doesn't make any sense to me. Conceptually, ALL sampled signals (even sine waves!) have "infinite bandwidth" until you apply the reconstruction/anti-imaging filter. White noise simply contains all frequencies, which allows you to easily visualize the reconstruction filter's response through the Nyquist frequency and beyond (which consists of images of the spectrum below Nyquist). You could also find the response of the reconstruction filter beyond Nyquist using sine waves by looking at the images.

You have skipped over #1, created an artificial signal that by definition, has infinite bandwidth. Here is the Fourier transform of an impulse/delta function:

That's the Fourier transform of an ideal, continuous time delta function (i.e. the Dirac delta function). Assuming a perfect reconstruction filter, a band-limited, discrete-time delta function is the same as the impulse response of said reconstruction filter: the normalized sinc function. If the peak is aligned to a sampling instant, you get the Kronecker delta function.

 

[amirm]

Perhaps I'm missing something, but what you seem to be claiming doesn't make any sense to me. Conceptually, ALL sampled signals (even sine waves!) have "infinite bandwidth" until you apply the reconstruction/anti-imaging filter.

What? The whole purpose of ADC filter is to make sure this doesn't happen. The source as such is definitely bandlimited or the system breaks. A sampled sine wave below Fs/2 most definitely does NOT have infinite bandwidth.

BTW, little in signal processing "makes sense." Do not try to intuit what signal processing is. It is not an analog concept to apply one's intuition to it.

You could also find the response of the reconstruction filter beyond Nyquist using sine waves by looking at the images.

There are two parts to reconstruction: generating a waveform and then filtering it. Of course if you don't filter it, you will have out of band images. But that is an unrelated thing to having out of band signals to begin with. In that case, even perfect filtering will not preserve that signal. As I explained, the theorem demands two filters, not one. One at capture and one at playback.

Assuming a perfect reconstruction filter, a band-limited, discrete-time delta function is the same as the impulse response of said reconstruction filter: the normalized sinc function.

A band limited impulse is no longer an impulse. The act of low pass filtering will cause it to become the sinc function which is not an impulse. What you state would then become sinc function of a sinc function.

Remember, sticking a one in a series of zeros does NOT create a band limited signal as no attempt was made to make it so. Relying on anti-image filtering in the DAC is improper as that would say that your system can reproduce infinite bandwidth which it clearly cannot.

 

[bmc0]

A sampled sine wave below Fs/2 most definitely does NOT have infinite bandwidth.

Try it. Play a perfect sine wave using some DAC at, say, Fs/2 - 1kHz, and then measure the output. You will see an image at Fs/2 + 1kHz at a level that corresponds to the reconstruction filter's response.

Do you not try to intuit what signal processing is. It is not an analog concept to apply one's intuition to it.

I have a reasonable understanding of the mathematics. I'm not trying to "intuit" anything.

Remember, sticking a one in a series of zeros does NOT create a band limited signal as no attempt was made to make it so.

Quoting the Wikipedia page for the Kronecker delta function:

[...] if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered (with cutoff at the critical frequency) per the Nyquist–Shannon sampling theorem, the resulting discrete-time signal will be a Kronecker delta function.

 

[amirm]

Try it. Play a perfect sine wave using some DAC at, say, Fs/2 - 1kHz, and then measure the output. You will see an image at Fs/2 + 1kHz at a level that corresponds to the reconstruction filter's response.

Did you not read what I explained? The imperfect response of the DAC filter will result in some imaging but that does not remotely indicate that the sine wave has infinite bandwidth.

 

[amirm]

Quoting the Wikipedia page for the Kronecker delta function:

This is the quote:

"Under certain conditions, the Kronecker delta can arise from sampling a Dirac delta function. For example, if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered (with cutoff at the critical frequency) per the Nyquist–Shannon sampling theorem, the resulting discrete-time signal will be a Kronecker delta function."

There was no filtering when the sequence of zero and one were created. Ideally or otherwise.

If said impulse needs such ideal filtering, then by definition has far more bandwidth as is.

I have a reasonable understanding of the mathematics. I'm not trying to "intuit" anything.

You are making comments that violate the most basic principles of sampling and signal processing. What math you know must be unrelated to these topics.