Time domain and frequency domain relation and measurements

[pma]

I am not sure that the relations between time domain and frequency domain signal representation are generally well understood, And I am also not sure that relation between transfer function, impulse response, step response and frequency responses (amplitude and phase) are well understood as well. So please let me ask a simple question - below is a measurement of amplifier response to a single input unipolar square of some 130 ms length. Is it clear that it is a response of the amplifier with amplitude frequency response of 3Hz - 100kHz (-3dB)?

Please kindly vote in the poll.

In case of any questions on time domain and frequency domain relation, or impulse and step responses, this is the right thread.

 

[pjug]

It is clear that it is high pass filtered. I am not dialed in enough to know whether a 3Hz filter gives this exact response without simulating. So I would say no, for me it is not immediately clear to me just looking at the plot that the response matches the filter.

 

[Extreme_Boky]

I can see the amp is not DC-coupled. It seems that it is slew-limited as well, but where that limit is coming from (input and/or gain setting filters, or the gain stage linearity) - that is impossible to say. Any attempt I made to figure out the frequency bandwidth, based on slew rate, was unsuccessful because you used an unipolar square wave pulse... Anyway, I am sure you'll get a lot of replies. My vote: "I do not know"...

 

[pma]

I can see the amp is not DC-coupled. It seems that it is slew-limited as well, but where that limit is coming from (input and/or gain setting filters, or the gain stage linearity) - that is impossible to say.

That's correct, the amp is not Dc coupled, as most of power amplifiers. The slew rate limitation cannot be told from the posted because the time scale is 100ms per division. It could be told from time scale 10us per division, which would show slew rate, however not the low frequency limit.

An example of DC coupled amplifier is Purifi 1ET400A (low gain setting):

It transfers the unipolar long rectangle (250ms), as a proof it works from DC. It is a double bladed sword, because the DC protection acts no sooner than at 12Vdc output. On the other hand, such amplifier introduces no phase change at very low frequencies. Which may be an advantage.

 

[pma]

It is clear that it is high pass filtered. I am not dialed in enough to know whether a 3Hz filter gives this exact response without simulating. So I would say no, for me it is not immediately clear to me just looking at the plot that the response matches the filter.

Yes, it is not easy to tell. It is a second order high pass (thus the visible slow undershoot), probably not audible in most cases.

 

[DVDdoug]

I am not sure that the relations between time domain and frequency domain signal representation are generally well understood

Generally no... Probably less than 1% of the population understands the math and I'm NOT in that 1%.

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.

Frequency response (over the audible bandwidth) tells you everything you need to know. Our ears are (obviously) bandwidth limited and time-limited (we can't hear the frequency/pitch of a 1-cycle 1kHz tone).

And we can't hear normal-reasonable phase shifts unless there are two identical simultaneous signals with different phase-shifts. There is something called an all-pass filter that creates different phase shifts at different frequencies. It completely fouls-up a square wave or pulse but it doesn't change the sound.

Vinyl cutting and playback does that too, or if you record a square wave from a speaker, especially a 2-way or 3-way with a crossover, a square wave gets all messed-up, both by bandwidth limiting and phase shifting.

 

[tmtomh]

The poll itself is not entirely clear, so I would suggest not inferring too much from the proportion of No votes to the proportion of I Don't Know votes. (The meaning of Yes votes is of course totally clear.)

The reason I say this is because if the individual cannot tell from looking at the graph because they themselves lack the understanding, they might vote either No or I Don't Know. Conversely, if the individual can tell, and they can definitively tell that it's not possible to see this just from that graph, then they might vote No.

In other words, a No vote could mean, "I know it's not possible to tell with certainty" or it could mean "I personally can't tell."

I presume the intent is for the I Don't Know response to cover all cases where the individual just doesn't know enough (and so that's what I voted). But people don't always read things super-carefully, especially when they are somewhat ambiguous like this.

At any rate, I appreciate the poll and topic because it's an interesting subject - and at the same time the graph simply could have been used to explain or demonstrate the relationship between frequency response and phase response. But as usual @pma has a deep need to demonstrate his superior expertise over the majority of the membership here, whose intelligence and knowledge level he loves to insult.

 

[Grotti]

I know a lot more about the physics of loudspeakers and I had to admit in the poll, that I could not figure out the answer from the graph. But really, Pavel: what is the purpose of this thread? Instead of showing the majority here what they don't know, you could have explained the "riddle" right from the start. Everyone knows about your deep insight in the physics of amplifiers: why don't you share your wisdom and explain how to read the graph and draw the right conclusions? The poll is useless in my opinion.

 

[IPunchCholla]

below is a measurement of amplifier response (red) to a single input unipolar square of some 130 ms length (blue).

Can you determine the filter applied and bandwidth of the amp?

Is that an accurate rendering of the question? Because I had a real hard time even determining what I was being asked.

 

[pma]

Can you determine the filter applied and bandwidth of the amp?

Is that an accurate rendering of the question? Because I had a real hard time even determining what I was being asked.

Sorry if I was not clear enough. The amplifier used in the test question has amplitude and phase frequency responses as shown below:



My question was/is if it is clear to readers that such amplifier has a response to 130 ms (or similar) length unipolar rectangular impulse looking like this:



it is in fact a question on awareness of direct relation between frequency domain and time domain responses. The usefulness of time domain tests is being doubted here on the forum, and I think that is probably because of lack of awareness of such relationships between time and frequency domain. Though the signal used may be called as "unnatural" in music, there exist music samples with high asymmetry and these asymmetric signals may quite easily send the amplifier to clipping or a speaker to high cone excursion, depending on the signal shape. This is prevented if the amplifier bandwidth is extended to DC, like with the Ncore and Purifi amplifiers. However, DC protection circuit design and setting is a challenge then.

 

[Sokel]

I know by chance, just because I have seen it measuring an amp with the exact low freq cut, my silly measurements with various el. crossovers and by testing Multitone Analyzer.

First time one measures an x-over and looking at the resulting waveform is scary, I thought something was broken.
So I learned by necessity

 

[KSTR]

Is it clear that it is a response of the amplifier with amplitude frequency response of 3Hz - 100kHz (-3dB)?

Well, there is an engineering rule-of-thumb that a first order highpass' step response has dropped in half in about 1/10 of the period of the filter frequency, and for second order thus the 1/10th period point is roughly when the roof is down at 25%. I would eye-ball this to ~40ms from your plot, thus period = ~400ms ==> 2.5Hz.

The 100kHz upper limit is not visible, but one could eye-ball from the plot that there are about like 100 pixels in a 100ms span, so the edge sure is faster than ~1ms which points to a high-frequency cutoff at higher than 10kHz, roughly.

 

[tmtomh]

Sorry if I was not clear enough. The amplifier used in the test question has amplitude and phase frequency responses as shown below:

View attachment 469408

My question was/is if it is clear to readers that such amplifier has a response to 130 ms (or similar) length unipolar rectangular impulse looking like this:

View attachment 469409

it is in fact a question on awareness of direct relation between frequency domain and time domain responses. The usefulness of time domain tests is being doubted here on the forum, and I think that is probably because of lack of awareness of such relationships between time and frequency domain. Though the signal used may be called as "unnatural" in music, there exist music samples with high asymmetry and these asymmetric signals may quite easily send the amplifier to clipping or a speaker to high cone excursion, depending on the signal shape. This is prevented if the amplifier bandwidth is extended to DC, like with the Ncore and Purifi amplifiers. However, DC protection circuit design and setting is a challenge then.

Let me first say I do not fully understand the relationship between frequency and time-domain responses, so I appreciate you raising the subject, and I hope to learn from the thread (as I do from many ASR threads).

That said, it would be a lot easier if you would clarify some key aspects and stop burying the inevitable argument that's under the surface and instead just come out with it from the outset.

Here are some questions that would be very helpful to me. And while I know you and I often do not get along here, I assume I am not the only one with these questions and so I would hope you'd still want to answer or clarify them:

1. I understand the amplifier used in the test has the amplitude and phase responses you have shown in your graphs. With regard to the top graph in the post of yours I've quoted here, are you saying that this amplifier happens to have those responses? Or are you saying that any amplifier with the frequency response as depicted by the top blue line will have phase response as depicted by the bottom orange line? Again referring to your top graph, I see that the the ultrasonic phase and amplitude nonlinearities are roughly parallel, while the infrasonic (and to a small degree, bottom-audible-octave) phase and amplitude nonlinearities are roughly inverse of each other. I understand we're comparing amplitude and time/angle so perhaps the directions don't matter - but the relationship between the two curves is roughly opposite at each end of the spectrum - so is that an intrinsic aspect of the phase-amplitude relationship, or an aspect that just happens to be the case with this amp?

2. If you have to use an unnatural signal to show this stuff, but similar signals exist in real music, why are such real musical signals never used, in order to help demonstrate that this really does matter?

3. When the "usefulness of time domain tests is questioned" here at ASR, are people specifically saying that what you're showing in your graphs is not an issue? And is the magnitude of what you show in your graph sufficient to actually be an issue?

Thank you for any answers you can provide.

 

[IPunchCholla]

Sorry if I was not clear enough. The amplifier used in the test question has amplitude and phase frequency responses as shown below:

View attachment 469408

My question was/is if it is clear to readers that such amplifier has a response to 130 ms (or similar) length unipolar rectangular impulse looking like this:

View attachment 469409

it is in fact a question on awareness of direct relation between frequency domain and time domain responses. The usefulness of time domain tests is being doubted here on the forum, and I think that is probably because of lack of awareness of such relationships between time and frequency domain. Though the signal used may be called as "unnatural" in music, there exist music samples with high asymmetry and these asymmetric signals may quite easily send the amplifier to clipping or a speaker to high cone excursion, depending on the signal shape. This is prevented if the amplifier bandwidth is extended to DC, like with the Ncore and Purifi amplifiers. However, DC protection circuit design and setting is a challenge then.

Thank you for the clarification. But I still don’t know how to answer the question. Looking at the impulse response, I can see that the amplifier is not flat to zero and tails off at the high end of the bandwidth as well. But I can’t tell by how much, nor can I tell the bandwidth. What should be my answer to the question?

 

[Keith_W]

I am not sure that the relations between time domain and frequency domain signal representation are generally well understood, And I am also not sure that relation between transfer function, impulse response, step response and frequency responses (amplitude and phase) are well understood as well. So please let me ask a simple question - below is a measurement of amplifier response to a single input unipolar square of some 130 ms length. Is it clear that it is a response of the amplifier with amplitude frequency response of 3Hz - 100kHz (-3dB)?

Please kindly vote in the poll.

View attachment 469356

In case of any questions on time domain and frequency domain relation, or impulse and step responses, this is the right thread.

Thanks for your response. Can you please explain how you can derive that an amplifier has an amplitude response from 3Hz - 100kHz with a -3dB response at the top end from this step response alone?

 

[pma]

Thanks for your response. Can you please explain how you can derive that an amplifier has an amplitude response from 3Hz - 100kHz with a -3dB response at the top end from this step response alone?

First, the plot posted is not the step response. It is a response to a rectangular impulse of 130 ms length. Step response is a response to a unity step. Unity step is 0 for t<0 and 1 for t>0. Such step response for the amplifier under test looks like this:

Now we are talking about low frequency (-3dB) corner. Amp response to unity step is the red line. Draw a tangential line and read the crossing with X axis. It is at 44ms. Suppose a simple RC highpass filter. 44ms would be a time constant Tau = RC. Calculate -3dB corner from Tau as Fd = 1/(2*pi*Tau). You get 3.6Hz and this is what I really measured, so the RC approximation was sufficient.

Now the high frequency corner. You need to use measurement with much shorter time base, like below:

This is a rising edge of the step response and you may use a square wave generator. Read time from 10% to 90% amplitude, it is called step response rise time 10% - 90%. If you solve equation for an RC lowpass filter (1 - exp(-t/Tau)) for 10% and 90% points, you get simple formula for high frequency corner as Fh = 0.35/Tr. It is again an approximation, however quite a good one.

----------------

If you want to go precisely, then:

1) measure step response with fast sampling and store the data. You get step response g(t)
2) make a derivative of g(t) to get impulse response h(t), h(t) = d(g(t))/d(t)
3) from known impulse response h(t) you get amplitude and phase responses via Fourier transform

 

[pma]

Different views of the LF behaviour:

 

[KSTR]

though the signal used may be called as "unnatural" in music, there exist music samples with high asymmetry and these asymmetric signals may quite easily send the amplifier to clipping or a speaker to high cone excursion, depending on the signal shape. This is prevented if the amplifier bandwidth is extended to DC, like with the Ncore and Purifi amplifiers.

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.

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

 

[Keith_W]

Hmm. I created a 50Hz square wave (red) and two minphase bandpass filters - 20Hz - 20kHz, and 20Hz - 1kHz (upper graph, showing amplitude response).

In the lower graph, we have the original square wave, along with the output of the 20-20kHz filter (blue) and the 20-1kHz filter (pink).

I have been staring at the graph trying to work out how to calculate the frequencies of the bandpass like you described but i'm kind of stuck. But never mind, I make a living from selling drugs and not from signal processing so I don't really need to know. All I need to know is that you can derive one from the other.

 

[amirm]

Sorry if I was not clear enough. The amplifier used in the test question has amplitude and phase frequency responses as shown below:

So instead of showing this measurement which everyone understands and can contain far more information, you decided that people should be able to reverse engineer a square wave response? It is like suggesting we get rid of English language here and use morse code instead.

Practice of square wave testing dates decades back when people didn't have ready access to audio analysis hardware and software. It was a highly crude method to deduct something about the nature of (usually) amplifiers. Today, there is no reason for anyone here to be concerned about square wave response of a system. I have explained this in a video I produced a while back: