Time domain and frequency domain relation and measurements

[IPunchCholla]

The artefact (peak) is a result of the blue scale (input, 0.6V) and red scale (amp output, 33V)) not matching exactly the gain of the amp.
The downward slope is a result of the second order high pass filter with 3.6Hz corner frequency that @pma described.
The overshoot is (another) result of this filter.
The downward peak is asymetrical relative to the upward one, because the "overshoot" is still negativ in the moment the downward step happens.

Thank you so much!

The artefact (peak) is a result of the blue scale (input, 0.6V) and red scale (amp output, 33V)) not matching exactly the gain of the amp.

If they were scaled exactly, the red would be slightly below the below the blue showing the high frequency fall off? I guess I’m confused on this because I am not seeing the treble cut shown in the FR in the impulse as was shown in the linked article that @levimax posted.

The downward peak is asymetrical relative to the upward one, because the "overshoot" is still negativ in the moment the downward step happens.

If the square wave was bipolar and repeated the second and third peaks of the response would have equal magnitude?

 

[olieb]

If they were scaled exactly, the red would be slightly below the below the blue showing the high frequency fall off? I guess I’m confused on this because I am not seeing the treble cut shown in the FR in the impulse as was shown in the linked article that @levimax posted.

What happens at the peaks can not be seen in the graph as this is much to fast for the time axis scale.
It is a bit more complicated than I mentioned above as the LP filter (100kHz) has an overshoot, too.
I made the step (not impulse) response for a FR with high pass 3.6Hz and low pass 10kHz (as I could not make REW create 100kHz filter).
On the left is the time scale similar to the diagram from OP and one can see all the effects of the high pass (3.6Hz) mentioned. There is a downward slope (not horizontal) and an (negative) overshoot.

On the right I used a much magnified time scale and one can see what the low pass (10kHz) does. There is a finite slope (not vertical) and an overshoot. With 100kHz this would be 10 times shorter of course.
From the 10kHz filter only the "blip" in the peak remains.

If the square wave was bipolar and repeated the second and third peaks of the response would have equal magnitude?

Maybe it will take a bit longer but eventually the output will get symmetrical as the DC part gets filtered out.

 

[KSTR]

This is a useful insight, but I’d expect it to be a routine problem with a well-established solution? Yes, a standard fixed window FFT will inevitably smooth over such bursts, and it’s not the ideal tool for real-time analysis in any case. I would think there should be a commonly available tool that, for any frequency band - regardless of its bandwidth - can accurately capture the true maximum total amplitude and the maximum slew rate observed, whether the content is broadband or extremely narrow, and display it as part of the spectrum. In other words, something that shows, for each band, the actual peak demands placed on the system, no matter whether the burst lasts only a few cycles or extends over many.

There is an analog solution available for decades, an analog filter bank followed by peak metering, sometimes even integrated into power amps:

Circuit is quite simple, a bunch of 2nd order bandpass filters followed by very rudimentary peak metering (only positive wave is used, etc), then driving a bar graph from the output levels of those peak meters.

Can be done in DSP these days, key is to use simple IIR bandpass filters not too narrow and not with too steep slopes as that would deform the waveforms too much, compromising the peak meter value readings. The bar graph spectrum meter of RME's ADI-2 Pro, DAC and 2/4 is made this way, sadly there is no infinite peak-hold option.

 

[danadam]

I would think there should be a commonly available tool that, for any frequency band - regardless of its bandwidth - can accurately capture the true maximum total amplitude and the maximum slew rate observed, whether the content is broadband or extremely narrow, and display it as part of the spectrum. In other words, something that shows, for each band, the actual peak demands placed on the system, no matter whether the burst lasts only a few cycles or extends over many.

Isn't "Peak" trace in REW's RTA something like that?

 

[tengiz]

There is an analog solution available for decades, an analog filter bank followed by peak metering, sometimes even integrated into power amps:
View attachment 469858

Circuit is quite simple, a bunch of 2nd order bandpass filters followed by very rudimentary peak metering (only positive wave is used, etc), then driving a bar graph from the output levels of those peak meters.

Can be done in DSP these days, key is to use simple IIR bandpass filters not too narrow and not with too steep slopes as that would deform the waveforms too much, compromising the peak meter value readings. The bar graph spectrum meter of RME's ADI-2 Pro, DAC and 2/4 is made this way, sadly there is no infinite peak-hold option.

Yes, something like that! It should be trivial to implement it in DSP even with modest hardware. Even simpler for the use case that you brought up - the multiway speaker design. You only need so many frequency band bins for speakers. Implementing a true peak meter, I mean.

 

[tengiz]

Isn't "Peak" trace in REW's RTA something like that?

View attachment 469872

REW’s RTA appears to be based on FFT, which means it can miss some things.

A short sine burst and a continuous sine of the same frequency and amplitude have the same instantaneous power, so either can momentarily stress your amp equally. But in the FFT, in addition to not accumulating the power per bin for the entire duration of the FFT window, the burst’s energy is also spread across many frequencies, further lowering the power per bin.

To be specific, I ran a quick test:

With the FFT bin spacing matched to the signal frequency (4000.49 Hz at 48 kHz sample rate, 32 k FFT), a continuous 1 V-peak sine wave puts essentially all of its spectral content into a single FFT bin, producing a bin magnitude of about 1.00 V. The same tone played as a 100-cycle burst still reaches 1 V peak in the time domain shows the main bin at only 0.0366 V. This ~27× difference shows how an FFT can make a short, high-amplitude burst look much weaker at its nominal frequency, even though it can stress an amplifier just as much instantaneously as a sustained tone.

 

[levimax]

REW’s RTA appears to be based on FFT, which means it can miss some things.

A short sine burst and a continuous sine of the same frequency and amplitude have the same instantaneous power, so either can momentarily stress your amp equally. But in the FFT, in addition to not accumulating the power per bin for the entire duration of the FFT window, the burst’s energy is also spread across many frequencies, further lowering the power per bin.

To be specific, I ran a quick test:

With the FFT bin spacing matched to the signal frequency (4000.49 Hz at 48 kHz sample rate, 32 k FFT), a continuous 1 V-peak sine wave puts essentially all of its spectral content into a single FFT bin, producing a bin magnitude of about 1.00 V. The same tone played as a 100-cycle burst still reaches 1 V peak in the time domain shows the main bin at only 0.0366 V. This ~27× difference shows how an FFT can make a short, high-amplitude burst look much weaker at its nominal frequency, even though it can stress an amplifier just as much instantaneously as a sustained tone.

Thank you, I learned something, I was under the impression that FFT was more or less "perfect math".

 

[tengiz]

Thank you, I learned something, I was under the impression that FFT was more or less "perfect math".

We all learn something every day - it's good to be addicted to fun in learning

It is perfect math - but only for the entire block of samples the FFT is computed on, so there’s an inherent averaging/normalization. If a frequency component is present for only part of that block (like a burst), the FFT will show it attenuated, with the amount of attenuation depending on the ratio of the burst length to the FFT block length.

 

[pma]

To put the debate on some solid ground, please let me post some basic terms of system responses and signal analysis, from respectable sources (citation at the end of the post):
------------------------------

Signal types
The most fundamental division is into stationary and non-stationary signals. It is sufficient to interpret stationary signals as those whose average properties do not vary with time and are thus independent of the particular sample record. This applies to both deterministic and random signals.

The instantaneous value of a stationary deterministic signal is predictable at all points in time, while with stationary random signals it is only the statistical properties which are known.

Non-stationary signals may be roughly divided into continuous non-stationary signals (of which a good example is speech) and transient signals which may be defined as those which start and finish at zero. The difference is fundamentally that a transient signal is analyzed as a whole, whereas a continuous non-stationary signal, such as speech, will be analyzed in short sections, each of which will often be quasi-stationary.

Glossary

Delta function: A normalized impulse. The
discrete delta function is a signal composed of all
zeros, except the sample at zero that has a value of
one. The continuous delta function is similar, but
more abstract.

Discrete time Fourier transform (DTFT):
Member of the Fourier transform family dealing
with time domain signals that are discrete and
aperiodic.

Fast Fourier transform (FFT): An efficient
algorithm for calculating the discrete Fourier
transform (DFT). Reduces the execution time by
hundreds in some cases.

Time domain: A signal having time as the
independent variable. Also used as a general
reference to any domain the data is acquired in.

Frequency domain: A signal having frequency as
the independent variable. The output of the
Fourier transform.

Impulse: A signal composed of all zeros except
for a very brief pulse. For discrete signals, the
pulse consists of a single nonzero sample. For
continuous signals, the width of the pulse must be
much shorter than the inherent response of any
system the signal is used with.

Impulse response: The output of a system when
the input is a normalized impulse (a delta
function).

Nyquist frequency, Nyquist rate: These terms
refer to the sampling theorem, but are used in
different ways by different authors. They can be
used to mean four different things: the highest
frequency contained in a signal, twice this
frequency, the sampling rate, or one-half the
sampling rate.

Sampling theorem: If a continuous signal
composed of frequencies less than f is sampled at
2f , all of the information contained in the
continuous signal will be present in the sampled
signal. Frequently called the Shannon sampling
theorem or the Nyquist sampling theorem.

Step response: The output of a system when the
input is a step function.

Literature:

[1] Steven W. Smith: The Scientist and Engineer's Guide to Digital Signal Processing, Second Edition.

[2] R. B. Randall, Application of B&K Equipment to Frequency Analysis.

P.S.: Please note that: Time domain: A signal having time as the independent variable. Also used as a general reference to any domain the data is acquired in.

 

[pma]

Below is a DAC output frequency spectrum (Topping D10s) to a single, non-zero digital domain delta impulse, for 44.1kHz sampling rate. Digitized by E1DA Cosmos ADC with 192kHz sampling frequency.

1. Signal (samples) in digital domain (the shape is drawn by sin(x)/x display reconstruction):

and the DAC analog output amplitude frequency spectrum: