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

[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.

Making sure everyone understands where he is going wrong, this is the process:

Focusing on the third graph, the discrete nature of digital to analog conversion creates new, high frequency content due to those sharp edges/steps. Filtering removes those, resulting in our original signal, devoid of these artifacts. @bmc0 is claiming because these artifacts remain to small extent due to less than ideal DAC filtering, it proves that the original signal had them as well and hence, has infinite bandwidth. Hopefully you all see why this is a completely wrong conclusion as that high frequency content was introduced in the DAC and did not exist in the original signal (first graph).

 

[Holmz]

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.

I feel your pain.
But is it BW < Fs/2 or BW <= Fs/2 ?

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.

I know - but that math part seems to be ignored in favour of legalities.

… 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:

View attachment 469829

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

View attachment 469830

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.

It being grasped upon as defensible in the context of same DUT, but those are also manufactured and designed using DSP… So we have a disconnect somewhere.

Did you not read what I explained? The imperfect response of the DAC filter will result in some imaging

I assume that should have been “ringing”?
(And got spell checked.)

but that does not remotely indicate that the sine wave has infinite bandwidth.

Correct, it cannot have a frequency greater than Nyquist.

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.

100%

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

Isn’t it already limited by Nyquist to be only “up to” the Nyquist frequency?

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:

I think if you skip step #1, then it does not contain frequencies to infinity…
However it does alias all the frequencies by folding them back onto/into the bandwidth of the sample rate?

 

[Holmz]

We sort of delved down the rabbit hole of Theory…

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??

It would be pretty easy to test it with some corruption of the digital data.

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.

Having some errors in transmission happens, but there are tons of error correction schemes.
But knowing how the system responds to bit errors, seems like a valid thing to do.

 

[bmc0]

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

Did you not read the quote correctly? A Dirac delta function which is aligned to a sampling instant becomes a Kronecker delta function after ideal bandlimiting and sampling. A Kronecker delta function is exactly the sequence you claim not to be bandlimited.

@bmc0 is claiming because these artifacts remain to small extent due to less than ideal DAC filtering, it proves that the original signal had them as well and hence, has infinite bandwidth.

No. You are misrepresenting what I said. I did not say that the original, continuous time sine wave has infinite bandwidth. A discrete time signal is conceptually a sequence of shifted and scaled Dirac delta functions when represented in continuous time. This creates an infinite series of images above Fs/2 (assuming uniform sampling). The reconstruction filter removes these images (imperfectly in practice as an ideal filter is not realizable).

 

[Holmz]

Did you not read the quote correctly? A Dirac delta function which is aligned to a sampling instant becomes a Kronecker delta function after ideal bandlimiting and sampling. A Kronecker delta function is exactly the sequence you claim not to be bandlimited.


No. You are misrepresenting what I said. I did not say that the original, continuous time sine wave has infinite bandwidth. A discrete time signal is conceptually a sequence of shifted and scaled Dirac delta functions when represented in continuous time. This creates an infinite series of images above Fs/2 (assuming uniform sampling). The reconstruction filter removes these images (imperfectly in practice as an ideal filter is not realizable).

Ah … I think I see now that there needs to be a filter behind the DAC.
(Mathematically the FFT stopping at Nyquist is different.)
Thank you sir!

 

[NTK]

Did you not read the quote correctly? A Dirac delta function which is aligned to a sampling instant becomes a Kronecker delta function after ideal bandlimiting and sampling. A Kronecker delta function is exactly the sequence you claim not to be bandlimited.

You can only generate the single sample impulse (the Kronecker delta) by creating it in the samples domain, but you won't be able to get it from a real anti-alias filtered analog (continuous time) signal. See this post by J J.

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

Aug 14, 2025

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...

• j_j

• 

 

[bmc0]

You can only generate the single sample impulse (the Kronecker delta) by creating it in the samples domain, but you won't be able to get it from a real anti-alias filtered analog (continuous time) signal.

In practice, yes. But you can mathematically derive the discrete-time Kronecker delta function from a continuous-time Dirac delta function.

What if the ADC has a much sharper filter than the DAC? Is the discrete time signal from the ADC then "illegal"? The output from the DAC will contain content above Nyquist just like with the "artificial" signals. This is the main point I was taking issue with. A Kronecker delta function isn't "illegal" per the sampling theorem, but the sequence is infinite so it cannot be generated by a practical ADC (edit: unless the filter cutoff is at Fs/2, which is the case in some ADCs; this of course results in aliasing in practice as the filter cannot be ideal). How do you define what's valid in practice then? By how steep the DAC's reconstruction filter is? How can you do that if there's not a standard filter?

 

[amirm]

Did you not read the quote correctly? A Dirac delta function which is aligned to a sampling instant becomes a Kronecker delta function after ideal bandlimiting and sampling.

One more time: there is no band limiting applied when you create an impulse in digital domain out of thin air. Nor is there any "sampling." Where did you get that from? No analog signal was used to create the impulse. Such an analog signal does not exist in real world as it requires infinite energy. If it did, and you filtered it to have half the sampling rate, then it would not remotely look like an impulse. The math mandates it.

 

[tengiz]

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.

That would be the mirror image of the Fs/2 - 1kHz. It’s NOT in the sine wave itself - it’s an artifact of digital-to-analog conversion, essentially the inverse of ADC aliasing. Without a reconstruction filter, the DAC output contains not only the intended signal but also an infinite series of mirrored spectral images repeating at multiples of the sampling rate.

 

[MaxwellsEq]

A real-world ADC converts a real-world signal into a sequence of digits. To be a valid real-world ADC it MUST have a filter blocking everything above half the sampling frequency.

Consequently in the digital domain there are a set of number sequences which can NOT be created by a real-world ADC. We can easily create such number sequences using a computer. They may also be created by accident - corrupting captured sequences. Such corruptions can make splats and bangs, so checksums are used to protect playback.

Perhaps it's interesting to use such "non-captureable" number sequences to see what happens when they are fed through a DAC with its filter(s); e.g. we may learn about stability when handling corrupted files. But it has nothing to do with the reproduction of music which needs to have passed through an ADC with its filter.

 

[boxerfan88]

Focusing on the third graph, the discrete nature of digital to analog conversion creates new, high frequency content due to those sharp edges/steps.

This video is often quoted on ASR:

a sampled signal is entirely different it's discrete time it's only got a value right at each instantaneous sample point, and it's undefined -- there is no value at all everywhere in between

After D/A conversion, is it truly a "stair step"? Or not?
The voltage level after D/A conversion could be any value between the 2 sample points, correct?
It is not necessarily a straight line then a sudden jump up/down to the next sample like in the "stair step" picture.

Based on the picture below, it could have been the red trace, or green trace, or any other pattern so long as the D/A converted output (before filter) reaches the correct voltage at the sampling point. Is my understanding wrong?

After the filter gets rids of the HF junk, we should get back the original analog waveform. Yes / no?


/

 

[pma]

After D/A conversion, is it truly a "stair step"? Or not?

No. Only in case of zero-order hold (ZOH) filter, which is a simple type of reconstruction filter used in digital-to-analog conversion (DAC). It holds the value of a digital sample constant for the duration of the sampling interval, effectively creating a staircase-like output. If there was not this zero order filter, the output would be scaled delta functions (impulses). But such zero order filter introduces mirror images above Fs/2 and also 3.9 dB drop at Fs/2. So it is highly recommended to use oversampling digital reconstruction filter + simple analog low pass, to get acceptable output response.

 

[NTK]

In practice, yes. But you can mathematically derive the discrete-time Kronecker delta function from a continuous-time Dirac delta function.

What if the ADC has a much sharper filter than the DAC? Is the discrete time signal from the ADC then "illegal"? The output from the DAC will contain content above Nyquist just like with the "artificial" signals. This is the main point I was taking issue with. A Kronecker delta function isn't "illegal" per the sampling theorem, but the sequence is infinite so it cannot be generated by a practical ADC (edit: unless the filter cutoff is at Fs/2, which is the case in some ADCs; this of course results in aliasing in practice as the filter cannot be ideal). How do you define what's valid in practice then? By how steep the DAC's reconstruction filter is? How can you do that if there's not a standard filter?

That's before the samples get to the DAC, so what the DAC doess don't matter.

The only "legal" signal (i.e. that conforms to the bandwidth limitation requirements per the sample theorem) that will give these samples is the normalized sinc function. The sinc function is the impulse response of a perfect linear phase brickwall low pass filter with a cutoff frequency of fs/2 — a non-causal filter. Since we can't have non-casual filters in real life, the only method we'll have these samples (i.e. the Kronecker delta function) is by synthesizing them in the samples domain with a computer. You can't get these samples by anti-alias filtering an electrical impulse.

 

[pma]

The only "legal" signal (i.e. that conforms to the bandwidth limitation requirements per the sample theorem) that will give these samples is the normalized sinc function. The sinc function is the impulse response of a perfect linear phase brickwall low pass filter with a cutoff frequency of fs/2 — a non-causal filter. Since we can't have non-casual filters in real life, the only method we'll have these samples (i.e. the Kronecker delta function) is by synthesizing them in the samples domain with a computer. You can't get these samples by anti-alias filtering an electrical impulse.

Correct. And not understanding this properly has been a source of many misunderstandings in another threads.

 

[Albrecht47]

Bringing some content back to this thread:
https://www.audiosciencereview.com/...ith-major-audio-luminaries.62951/post-2374372
https://www.audiosciencereview.com/...ith-major-audio-luminaries.62951/post-2376889

With respect, we have audio luminary @j_j confirming his personal experience suggests that the title to OP should end with Yes.

Some posters in this thread seem to assign significant weight to J_J's statements and ,along with their own experience, agree that answer to the OP question is Yes.

But doesn't that contradict, again with respect, other audio luminaries @Floyd Toole , Lipshitz, Stanley P.; Pocock, Mark; Vanderkooy, John results from their large expensive tests, the likes of which J_J refers to?

Research has confirmed that phase distortions are audible. I personally find that very general statement to be of little use.

Research has also confirmed that the typical listener, in a typical living room, listening to typical source material and typical equipment, at an average typical SPL, with average and trained ears, will not be able to conclusively hear phase distortion.

As an audio nobody, I'm interested in the answer to the OP question as it relates to me listening to rock music on my hifi system in my living room. I don't care being able to hear a difference in an anechoic chamber with test tones. Even if I could hear a subtle difference in my environment, unless it can be scientifically proven ( rigorously controlled tests) to be preferable to a statistically significant # of listeners, I don't care.

In the context of our hifi hobby, based on the audio luminary rigorously controlled test results (no offense to J_J and his personal experiences), not based my own far very not controlled test results, I'm leaning in the phase isn't audible direction.

To address everyone's perspectives, and take into account all test results regardless of preferential relevance or rigor, perhaps the answer to the OP question should be "it depends"?

 

[j_j]

The only "legal" signal (i.e. that conforms to the bandwidth limitation requirements per the sample theorem) that will give these samples is the normalized sinc function.

Not really, when one applies what we know about absolute thresholds, physics thresholds, etc, then filters that do not go to zero outside the passband, but that are down 120dB or so, are easily shown to be satisfactory because the error is below the noise floor of the atmosphere at your ear drum.

In order to go to zero, the filter must be infinite in length. To be pure delay, it has to be symmetric about the middle of the impulse response, thereby making the acquisition have infinite delay.

That's a bit much in the real world, where we have some known physics that provide noise floors that are simply not going to go away.

For instance, 18 bits is pretty much sufficient in almost any real (not research space) listening room.

 

[j_j]

In the context of our hifi hobby, based on the audio luminary rigorously controlled test results (no offense to J_J and his personal experiences), not based my own far very not controlled test results, I'm leaning in the phase isn't audible direction.

Well, if a signal that demonstrates such audibility in a consumer living room does not suffice, well, ok, I suppose. You've being extremely misleading when you say "personal experience" here, since absolute demonstration of phase audibility is really quite incontrovertable. The question is down to "does it matter with most normal content", and that's where work needs to be done.

As to the various "rigorously controlled tests", I would suggest that you study, carefully, what they actually tested. You are, again, misleading the reader.

 

[bmc0]

Without a reconstruction filter, the DAC output contains not only the intended signal but also an infinite series of mirrored spectral images repeating at multiples of the sampling rate.

Yes, I stated this previously.

The only "legal" signal (i.e. that conforms to the bandwidth limitation requirements per the sample theorem) that will give these samples is the normalized sinc function.

You are essentially restating what I already said. Amir has repeatedly claimed that such a signal is not bandlimited, but, as we both explained, it is bandlimited. It is true that one cannot get exactly this signal from a continuous time impulse with a practical antialiasing filter unless the filter allows some aliasing[1], but this is not relevant to the point being made.

Consider that a practical DAC has a finite-length reconstruction filter. The Kronecker delta is infinite length, but only a length equal to twice that of the reconstruction filter matters to the DAC's output—the infinite zeros preceding and following have no bearing on the reconstructed output. Thus, the DAC's output given the infinite, perfectly bandlimited sequence must be identical to that given the truncated sequence. If one wants to claim that a truncated Kronecker delta is invalid, then the validity of a given sequence must be determined from the characteristics of the reconstruction filter. As there is no standard filter, this is impossible in the general case.

I previously mentioned a sine wave near Nyquist because it is a clear example of a signal that can be created using a realizable antialiasing filter which will also be reconstructed inaccurately by the filters found in most DACs. Take the D30Pro (posted previously) as an example: the filter is only ~9dB down at Fs/2, so any sine wave input sufficiently close to (but still below) Fs/2 will result in an easily measurable image above Fs/2. Does that mean that the original sine wave was not "properly bandlimited per the sampling theorem"? No, of course not.

Further, Amir's claim that a digitally-generated white noise signal contains frequencies above Fs/2 while a "properly bandlimited" signal does not is nonsensical for the same reason. The content above Fs/2 is due to images not removed by the DAC's reconstruction filter. One could, for example, generate a white noise sequence with Fs=44.1kHz which is bandlimited using a realizable low pass having a transition band that starts at, for example, 21kHz and ends at 22kHz. Now put it through the D30Pro with filter 2 (the steepest one) and look at the output. It will have content in the 23kHz-24kHz band just as with the other signal, but there will be a notch in the 21kHz-23kHz band. Is this input signal "properly bandlimited"? If not, how do you define "properly bandlimited"? If yes, then why is that not the case for the other white noise signal or for the Kronecker delta? The only mathematical difference is a steeper input filter.

[1]: Of course, you can (correctly) claim in this case that the input was not properly bandlimited, but this also reflects the reality for many ADC filters. For one example, see the AKM AK5578 datasheet (their top-of-the-line ADC). Page 11 shows the filter characteristics. Looking at figure 3, we have an equiripple filter with the cutoff at Fs/2—exactly the kind of filter which could produce a (truncated) Kronecker delta from a continuous time input.

 

[NTK]

Not really, when one applies what we know about absolute thresholds, physics thresholds, etc, then filters that do not go to zero outside the passband, but that are down 120dB or so, are easily shown to be satisfactory because the error is below the noise floor of the atmosphere at your ear drum.

In order to go to zero, the filter must be infinite in length. To be pure delay, it has to be symmetric about the middle of the impulse response, thereby making the acquisition have infinite delay.

That's a bit much in the real world, where we have some known physics that provide noise floors that are simply not going to go away.

For instance, 18 bits is pretty much sufficient in almost any real (not research space) listening room.

Agreed.

 

[pma]

Not really, when one applies what we know about absolute thresholds, physics thresholds, etc, then filters that do not go to zero outside the passband, but that are down 120dB or so, are easily shown to be satisfactory because the error is below the noise floor of the atmosphere at your ear drum.

I have not measured, yet, a real world DAC that would not have a kind of mirror image issues, spread as well below Fs/2, when the signal frequency approaches to Fs/2. Maybe the manufacturers and designers might add some margin and decrease filter Fc a bit to get rid of the mirror images garbage. The same applies to ADC anti-aliasing filters, which are often worse.

Below is an example, which is still very good. However not technically perfect.