Time domain and frequency domain relation and measurements

[LF78]

When I studied mathematic at university I was convinced (with proof) that frequency domain (amplitude + phase) and time domain are two equivalent representation of a signal, hence now I give this for granted

 

[danadam]

But I know enough that they are mathematically interchangeable (in theory). In the real-practical world with real-world audio the FFT conversion to frequency domain, and back, is imperfect. I believe it works with an infinitely-long continuous signal but not as well with audio that changes moment-to-moment.

Seemed to work here (unless I misunderstand what you mean):

Post in thread 'Master Thread: Are Measurements Everything or Nothing?'

Dec 28, 2022

Can you reproduce the original [music] waveform from its Fourier Transform?

Yes, seems to work for me.

Here's 30 sec of left channel of "Blågutten / Quiet Winter Night / Hoff Ensemble" from 24/96 file that was available on http://www.2l.no/hires/ (sadly, not anymore):

sox "2L-087_stereo-96kHz_06.flac" left.flac trim 0 30 remix 1

Using octave we can do FFT:

[X, Fs] = audioread("left.flac");
Y = fft(X);
save left_fft.txt Y;
exit

We get a bunch of complex numbers:

]$ head left_fft.txt
# Created by Octave 5.2.0, Wed Dec 28 20:03:53 2022 CET...

• danadam

• 

 

[boxerfan88]

I voted "No". After reading a few post ... I realized I forgot about DC-coupling requirement.
Silly me. Very good trick question!

 

[tengiz]

View attachment 469356

You can kinda eyeball the lower boundary from this picture, as already noted. But why does it overshoot?

 

[RayDunzl]

When I studied mathematic at university I was convinced (with proof) that frequency domain (amplitude + phase) and time domain are two equivalent representation of a signal, hence now I give this for granted

My old experiment:

Top: A 10Hz to 20kHz sine sweep through the speakers, into UMIK1, analyzed by REW, and presented as Step Response:

Bottom: A 10Hz square wave played through the speakers, into UMIK1, and recorded by Audacity, first 12ms of one cycle shown, acting as a "step".

 

[pma]

For a second-order highpass the headroom penalty is indeed ~7dB for the worst-case signal, a square pulse returning from +1 to -1 at the right moment when the undershoot is at its maximum. For a 3Hz highpass, that would be a square "base" frequency of 1/0.11s = 9Hz. Not that relevant, I'd say.
View attachment 469443

If the highpass corner frequency is 10Hz or even higher, actual clipping might occur occasionally where a DC-coupled amp wound not clip.

Sure, and that’s it and not easily imaginable immediately. Plus possible phase issues in case that low frequency corner is somewhere near 20Hz, like with many older amplifiers. IMO Bruno knew well why he had chosen the concept that starts from DC. Some DACs can do it as well, so it is possible to make an ABX test .

 

[pma]

When I studied mathematic at university I was convinced (with proof) that frequency domain (amplitude + phase) and time domain are two equivalent representation of a signal, hence now I give this for granted

Yes they are, but it may not be easy to realize what it means in the real world. To me, time domain is primary, we live in the 3D world + time, we live in a space-time and not in a space-frequency. Frequency domain was mathematically derived to make our view of certain phenomenae easier. Frequency domain is definitely secondary to time domain.

 

[amirm]

Frequency domain is definitely secondary to time domain.

Not if you want to understand audibility of impairments. Much of our knowledge of psychoacoustics is in frequency domain (naturally due to how our hearing works with auditory filter bank). To wit, lossy codecs use frequency transforms. Once there, they apply frequency masking to figure out the level of quantization that would be least audible as far as artifacts. Then (after a lossless compression), perform the reverse in the play to get your music back.

Just look at the dashboard view I show in all of my measurements:

The time domain response on top left is useless. The one in frequency domain on the right however, is massively informative.

Time domain analysis was useful in the analog recorder domain in measuring wow and flatter. Today, there is little to no use in time domain analysis.

 

[KSTR]

Frequency domain is definitely secondary to time domain.

My view is that both representations are equally important but have different use cases.

Just look at the dashboard view I show in all of my measurements:
[...]
The time domain response on top left is useless. The one in frequency domain on the right however, is massively informative.

There have been cases where we could identify a sample offset between channels of a DAC, directly visible in the waveform plot.

In general, though, for a THD+N test, the natural choice for the time-domain display would be the distortion+noise residual, preferably overlayed with the full output signal (scaled, of course). It often helps to interpret the distortion better.

For example, one can identify if the bulk of the distortion is just because the DUT is approaching saturation/clipping but would behave well at lower levels. OTOH, we could have crossover distortion -- where the performance would be really worse at lower levels -- which has its own distinctive residual pattern. These two distortion types may have very similar magnitude-only spectra at a high test level and thus cannot easily be identified.

 

[Extreme_Boky]

But why does it overshoot?

I'd say... contradicting effect (efforts) of the sound-coupling capacitor taking its time to reach zero after it was charged with that square wave rising edge, and the DC offset nulling circuit (still) doing its stuff at the moment the square wave pulse disappeared.

It'd be nice to see the schematic diagram... maybe later once we've been teased enough.

I like the thread... and I always appreciate the @pma posts.

 

[pma]

No teasing, the amplifier tested in post #1 has quite usual topology an usual frequency and time domain responses, with BW(-3dB) 3.6Hz - 98kHz when measured precisely.

Some time ago I have designed a special signal generator that generates 3 uniform rectangular impulses, like this:

First impulse length is about 200us, second impulse 120us, third impulse 45us approx. Rise time of the impulse edges is about 50ns (@7MHz -3dB BW), the length of the signal is sufficient to disclose excessive low frequency drop. The spectrum is flat to about 1kHz, then it continues to "infinity", as it is a transient signal that starts and finishes at zero and such signals have infinite spectrum.

The plot above shows amp response (red) to the test signal (blue) measured with high sampling rate. There is the amp rise time visible (and measured as 3.52us in the bottom left corner).
Above 100kHz we can see that the red trace deviates from the blue one as a result of amplifier 100kHz/-3dB high frequency corner.

When we make sampling with audio ADC, we get more or less distorted time domain plot (depending on sampling rate) and truncated spectrum, like below the result with Cosmos ADC and 44.1kHz sampling:


REW makes quite nice, but of course band-limited analysis

that is properly truncated below Fs/2, though the signal spectrum is in fact spread to infinity and in a real world would finish buried in the system noise somewhere above 10MHz.

Amplifier high frequency -3dB corner can be easily and more precisely measured when we increase time axis resolution and we then get Tr = 3.58us. F(-3dB) = 0.35/Tr.

 

[pma]

When we add a 2.2uF capacitor in parallel with the 4.7ohm load resistor, we get LRC circuit ringing at 80kHz:

This ringing is visible in both time and frequency domain, however frequency domain does not give a clear information on damping of the ringing and on percentage of the overshoot above the steady output value. Above 80kHz, the added capacitor creates an effective low-pass filter, this is on the contrary much better seen in the frequency domain than in the time domain. So both methods have their pros and cons and it is good to use both. Please also note increase of the rise time to 5.52 us (from 3.5us).

 

[Sokel]

For what it's worth I captured a 10Hz square and a 20Hz square as a whole system output, along with the room and every sin possible while my speakers have a F3 of 31Hz.
96KHz I/O, Band limited, 50% duty cycle.
Very strange results:



10Hz



20Hz

Poor thing tries to do 20Hz but all it does is a slight bumb.
The 10Hz is not visible at all.

(if someone is not used to it, squares sound scary! )

 

[tengiz]

I'd say... contradicting effect (efforts) of the sound-coupling capacitor taking its time to reach zero after it was charged with that square wave rising edge, and the DC offset nulling circuit (still) doing its stuff at the moment the square wave pulse disappeared.

It'd be nice to see the schematic diagram... maybe later once we've been teased enough.

Thank you. Makes sense - assuming the nulling circuit is separate from the main negative feedback and has a time constant similar to the input RC network. The impulse response with overshooting looked more like an active filter than an amplifier, which seemed odd to me.

I haven’t kept up with what’s “standard” in amp design for decades, but I do remember some amps with a dedicated DC servo just to hold the output offset at zero. I always thought that was a quirk of designs where the normal feedback couldn’t do it, or the input stage made it impractical.

 

[levimax]

Rod Elliott of Elliott Sound Products wrote a good article on using square waves to analyse amplifiers. Early in the article he shows that using a "real" square wave is not really useful as it can show issues that will never exist with music. Instead he uses a "bandwidth limited square wave" which still pushes the amp without being completely unrealistic. Here is the link with an excerpt from the article regarding bandwidth limited square wave below. https://sound-au.com/articles/squarewave.htm

When the input waveform is significantly faster than the amplifier stage, the leading and trailing edges will no longer be vertical, because the amplifying circuit has a limited bandwidth. It is very easy to perform a square wave test and end up with an entirely wrong answer if you're not careful. Much of the brouhaha that developed regarding TIM (transient intermodulation distortion) and/or SID (slew induced distortion) were due to the very fast risetime of the test signal. When testing any audio device, you must be aware of the simple fact that music does not contain very fast risetime signals, and most media (vinyl, CD, etc.) are actually not very demanding. This is because the amplitude of the musical harmonics is reduced by at least 6dB/octave from no higher than 2kHz or so. This means that the actual level at 20kHz will typically be 20dB lower than the level at midrange frequencies.

Therefore, an amplifier that can provide ±35V peaks will only be required to provide around ±3.5V peaks at 20kHz when operating just below full power with music as the input. This dramatically changes the required slew rate, but it's very common (and advisable) to ensure that an amplifier can reproduce no less than 50% output voltage at 20kHz to ensure an acceptable safety margin. TIM may have been discredited (along with its siblings), but it doesn't make any sense to limit an amplifier if it's not necessary. It also doesn't make sense to go to a great deal of additional effort to design an amplifier that can reproduce full power at 100kHz (or even 20kHz), because it will never be needed.

Most competent amplifiers can handle a band-limited squarewave with no fuss. Before using the squarewave, it should be passed through a filter that rolls off the response above 20kHz. Failure to use bandwidth limiting won't hurt the amplifier, but you may see artifacts that will not appear in normal use. A low-pass filter using a 1k resistor and 10nF capacitor gives a response that's considerably faster than the harmonic structure of music, but doesn't stress any amplifier too hard. The filter has a nominal -3dB frequency of 15.9kHz. My function generator has a risetime of 12ns for a 1V RMS squarewave - much too fast for even the most esoteric amplifier, so a filter is needed to prevent the DUT from slew rate limiting.

Figure 2 - Band Limited 1kHz Squarewave

The waveform shows the same squarewave seen in Figure 1A, but with a 1k + 10nF capacitor arranged as a low pass filter. So the waveform is easier to see, only four complete cycles are shown. This will be the case for all subsequent waveforms, and where possible the same vertical scale will be used as well. From this waveform, you can see the result of a low pass filter - the risetime is increased. As the input frequency approaches the filter's frequency, the effect becomes more obvious. Equally obvious is any circuit that applies high frequency boost, but we'll look at that in the next section.

It's also worth examining the risetime - it's usually measured between 10% and 90% of the waveform's peak-to-peak amplitude. The reason for this is simple, in that many circuits will have some small 'disturbance' as the voltage starts to change and just before it reaches the opposite peak voltage. By excluding the fist and last 10% of the waveform these disturbances are minimised and the true risetime (and from that the slew rate) can be determined more accurately.

 

[tengiz]

In general, this bickering on the forum about which specific kind of information is "more useful" or "more representative" puzzles me. Maybe I just don’t get it - I’m not in the trenches of the field’s current struggles and practices - but as a matter of common sense, doesn’t "the more solid data, the better" still hold true?? Sure, you need to prioritize so you don’t overwhelm customers, but between engineers, what’s wrong with giving multiple perspectives and angles when evaluating a piece of electronic gear - even if they overlap, or are essentially the same thing viewed from a different angle?

 

[levimax]

In general, this bickering on the forum about which specific kind of information is "more useful" or "more representative" puzzles me. Maybe I just don’t get it - I’m not in the trenches of the field’s current struggles and practices - but as a matter of common sense, doesn’t "the more solid data, the better" still hold true?? Sure, you need to prioritize so you don’t overwhelm customers, but between engineers, what’s wrong with giving multiple perspectives and angles when evaluating a piece of electronic gear - even if they overlap, or are essentially the same thing viewed from a different angle?

@amirm and @pma almost always "bicker" , and at least for me I learn a lot from the interaction.

 

[tmtomh]

In general, this bickering on the forum about which specific kind of information is "more useful" or "more representative" puzzles me. Maybe I just don’t get it - I’m not in the trenches of the field’s current struggles and practices - but as a matter of common sense, doesn’t "the more solid data, the better" still hold true?? Sure, you need to prioritize so you don’t overwhelm customers, but between engineers, what’s wrong with giving multiple perspectives and angles when evaluating a piece of electronic gear - even if they overlap, or are essentially the same thing viewed from a different angle?

Multiple perspectives are fine of course, but @levimax provided a great example above of the point @amirm was trying to make. Levimax wrote, “It is very easy to perform a square wave test and end up with an entirely wrong answer if you're not careful.”

That’s just an example, but the point is that “more tests, more robustness” is a good thing only to the extent that the additional and/or more demanding tests are designed and carried out in a way that avoids what Levimax describes.

 

[amirm]

I’m not in the trenches of the field’s current struggles and practices - but as a matter of common sense, doesn’t "the more solid data, the better" still hold true??

Sure. You don't see me complaining about members posting reviews with more/different measurements than I perform. The issue becomes the conclusions drawn from it:

Yes they are, but it may not be easy to realize what it means in the real world. To me, time domain is primary, we live in the 3D world + time, we live in a space-time and not in a space-frequency. Frequency domain was mathematically derived to make our view of certain phenomenae easier. Frequency domain is definitely secondary to time domain.

Our hearing system is a bank of auditory filters each tuned to a different frequency. Any text on psychoacoustics, will have a small portion on time domain aspects and the rest of the volume in frequency domain. If I don't comment and correct such posts pulled out of thin air, it will lead to people running off with wrong ideas about what makes a high performance piece of audio gear. Currently, high-end audio is full of these "time domain" claims used to sell any and all things. As such, it is critical to point out fallacies posted about them, especially in our forum.

 

[tmtomh]

Sure. You don't see me complaining about members posting reviews with more/different measurements than I perform. The issue becomes the conclusions drawn from it:


Our hearing system is a bank of auditory filters each tuned to a different frequency. Any text on psychoacoustics, will have a small portion on time domain aspects and the rest of the volume in frequency domain. If I don't comment and correct such posts pulled out of thin air, it will lead to people running off with wrong ideas about what makes a high performance piece of audio gear. Currently, high-end audio is full of these "time domain" claims used to sell any and all things. As such, it is critical to point out fallacies posted about them, especially in our forum.

And I’d add that so far @pma has not actually explained whether or not the measured response of the amp he used as an example could be anticipated to produce an audible effect when amplifying a musical signal, and if so, precisely on what basis would he be making such a claim.

Unless or until he does so, we’re left with the usual question in the audio hobby: an electrical and/or acoustic phenomenon is being referenced, but is it relevant and if so is it of a magnitude (and/or at a frequency) where it matters? The hobby has a lot of scientific hand-waving in which real phenomena are cited but their actual effects are never evaluated.

pma’s usual response to this kind of challenge is to either say we’re all too dumb to understand what he’s saying, or to accuse us of being “emotional” and therefore unwilling to consider his point. Hopefully he won’t fall back on either of those deflections here.