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T120 Klang kann man nicht messen? -2- — Transcript

by Physik der Elektrogitarre · 5,366 words · 840 segments · language en · Watch on YouTube

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0:09In the last video, mythology was given
0:11quite a bit of space. This time, it's
0:15more about signal analysis. An object
0:20creates sound, for example, a guitar
0:21amplifier. This sound is recorded by a
0:25microphone and then goes to analysis.
0:29In the past, that was a domain of
0:31analog technology. Today, one would
0:35probably do it all with digital
0:37technology, with digital signal
0:38processing. The first possibility to
0:44visualize such a signal is an
0:47oscilloscope. You look at the time
0:50function and already see, well, a few
0:54knobs. You can do a lot right, but also
0:57quite a bit wrong there. Sometimes the
1:00triggering isn't exactly easy,
1:02sometimes you also get images that are
1:03easy to interpret. But one must never
1:07forget, the ear is not an eye. The
1:11hearing has no way to directly analyze
1:14such a time function. In the inner ear,
1:19a spectral analysis occurs first, and
1:21we will now deal with that a bit more.
1:26On the topic of spectral analysis, I
1:30have several videos on the homepage.
1:35However, they are not generally
1:37accessible. I made these videos for
1:41company training sessions. If an
1:44employer says, we need this topic. If
1:48an employer spends money, then they can
1:51access these videos. They are not
1:55available for free. In the following,
1:58however, I will explain some images
2:00from these videos. A word of
2:02reassurance right now. It works without
2:04formulas. I will not go into all the
2:08details of these spectral
2:10transformations here. It looks
2:15complicated already. As I said, it
2:17works without formulas. You don't need
2:21much mathematical or physical knowledge
2:24here. The upper level, time functions.
2:29Time runs from left to right. A
2:32one-time event, a periodic event. That
2:36is the first essential distinction.
2:40When signals are to be analyzed, are
2:42they periodic in the time domain, that
2:45is, do they repeat or are they one-time
2:48events? Of course, you could now say,
2:51my god, life is finite and actually
2:54everything is a one-time event. There
2:56are certain rules. I will say several
2:59times that I cannot consider all the
3:01details here. The big distinction over
3:05here, time-discrete signals. The
3:09signals are sampled in the time domain
3:11and I think everyone knows that. When
3:15you work with a CD, you have a sampling
3:18frequency of 44.1 kHz, but there are
3:21also 48 kHz, 96, multiples of that, and
3:24other sampling frequencies. So either
3:29continuous-time or what is called
3:32time-discrete, and also in the discrete
3:36, there are one-time events and
3:39periodic events, time-periodic events.
3:45These distinctions are extremely
3:48important, because only when you know
3:51which of these signals you have can you
3:54also perform the correct spectral
3:57analysis. There are namely different
4:01transformations here and there are also
4:04different signal types in the frequency
4:08domain. Here at the front,
4:12frequency-continuous, one-time,
4:15frequency-periodic. That is perhaps an
4:19unexpected term now, but a spectrum can
4:22also repeat periodically and continue
4:24in the frequency domain. Back here, the
4:29frequency-discrete signals. When a
4:33computer or PC calculates a spectrum,
4:36it does not exist continuously. Instead
4:40, it exists at discrete frequency
4:42points. It is a bit unexpected that a
4:49frequency-periodic signal in the
4:52frequency domain does not also imply a
4:56time-periodic signal. I will mention
5:01several times why this is the case. I
5:03cannot explain it in detail. It is
5:07lecture material, but that would go
5:09beyond the scope here. There are four
5:13different transformations that connect
5:16the frequency function—the spectrum
5:19—with the time function. FT is the
5:23Fourier integral. How one writes these
5:28functions is not actually all that
5:29important. There are different
5:32notations used in textbooks as well. I
5:36always use round brackets for
5:37continuous functions and square
5:39brackets for discrete ones. But there
5:41are other ways to do it, too. When we
5:45connect both, we need the Fourier
5:47integral. Moving from the time domain
5:52to the frequency domain is done via
5:54analysis. When we reconstruct the time
5:59function from the spectrum, it is
6:01called synthesis or inverse
6:03transformation. For the two, we use the
6:08Fourier integral; for the two signal
6:10groups, the Fourier series. Here we
6:15have the discrete-time Fourier
6:17transform, and what is most commonly
6:20used here is the discrete Fourier
6:22transform, or DFT. And when performed
6:28with a fast computer algorithm, we call
6:31it the FFT, or Fast Fourier Transform.
6:36Usually, we are dealing with the DFT or
6:40FFT, and it connects these two signals
6:44or these two groups. And this is quite
6:49surprising, I think: the functions are
6:52periodic in both the time domain and
6:55the frequency domain, and it is not
6:58necessarily the period one expects when
7:01saying a tone is 1000 Hz, meaning a
7:04one-millisecond period. The periodicity
7:09comes from the analysis parameters.
7:13Both functions are discrete—
7:15time-discrete and frequency-discrete—
7:18and both functions are periodic—
7:21time-periodic and frequency-periodic.
7:26This results in peculiarities that can
7:29yield a result you did not expect in
7:32this form at all, which can potentially
7:36lead to total misinterpretation if you
7:39do not know which analysis parameters
7:42to set. In the following, I will point
7:47out some peculiarities and some
7:50parameter settings. I will not derive
7:53all the mathematical relationships in
7:56detail. That would go far too deep. It
7:59is therefore more of a warning: watch
8:02out, such transformations are not
8:04simple. One should acquire some
8:07technical knowledge if one does not
8:09already have it. A spectrum, as it is
8:14usually formed, is the sum of
8:16sinusoidal continuous tones. When we
8:22have the computer calculate the
8:24spectral function or spectrum for a
8:26time function, this spectrum shows us
8:29which continuous tones can be used to
8:32assemble such a signal. This is perhaps
8:37very surprising when you say, "Yes, but
8:39that was just a short pulse." How can
8:42it be composed of sinusoidal continuous
8:46tones? That is what the previous
8:49algorithm does, whether we want it to
8:51or not. If we do not want that, we must
8:53use a different algorithm. And one
8:57other thing is important, there is no
8:59spectrum for a single point in time.
9:03There is only a spectrum over a time
9:05interval. It is perhaps very tempting
9:09to say, I want a spectrum right now.
9:14However, this spectrum is always formed
9:16from a finite time interval. So not for
9:20a time duration of zero, but over a
9:23time interval of 5 milliseconds, 329
9:26milliseconds, or whatever parameters
9:29are set. If you let the time duration
9:35approach zero and say you only want the
9:38spectrum for that exact moment, then
9:40you get nonsense. Colloquially speaking
9:43, a few details, again without going
9:48too deeply into the mathematics. A time
9:54function, a single pulse in red, is
9:57sampled at different points in time,
10:00and only these samples reach the
10:03transformation algorithm. In the 1980s,
10:07there were expensive devices for this.
10:10A two-channel DFT analyzer back then
10:13cost 80,000 marks. Two years ago, in
10:17terms of purchasing power, that was
10:19about 73,000 €. Today, you would
10:22probably have to say one-to-one, 80,000
10:24€. The VW Beetle was an interesting
10:28car back then. For that money, you
10:31could have gotten eight VW Beetles,
10:33fully fueled. It was expensive. Today,
10:37you often get such DFT algorithms for
10:39free for PC use. Small problem there.
10:45Back then, you had cleanly defined,
10:48calibrated results. With the FFT
10:52programs often offered for free for PCs
10:55. You sometimes never know exactly how
10:58they are calibrated. For professional
11:02use, that is sometimes dangerous. Now
11:06for the windows, also only briefly,
11:09they are necessary. Up here, a time
11:14function in blue. Actually, this should
11:17be a sine function, sampled, black
11:20sample points, but a weighting window
11:23must be applied to it. A weighting
11:29window that pushes this sine function
11:31to zero at the beginning and end of the
11:33analysis. These windows have different
11:38names, e.g., the Hann window, which is
11:41named after Mr. von Hann, and this
11:43results in a spectrum that does not
11:46just sit in one spot; even though the
11:49analyzed signal is 1 kHz, this kind of
11:51spectrum comes out. Well, the spectrum
11:56is frequency-discrete and unfortunately
11:59has no line at this position, and
12:01therefore the spectrum spreads out.
12:04That is called leakage. And this
12:08leakage can be influenced, favorably or
12:11unfavorably, by the choice of the
12:13window. Okay, so we need windows, e.g.,
12:18for the Hann window, sometimes also
12:20called Hanning. Julius von Hann,
12:24Austrian meteorologist and physicist.
12:28Good, it is often used as a default
12:31window, and then people say, yes, that
12:34is set, so we measure everything with
12:36it. But there are also other windows.
12:40Uniform is also often called Dirichlet,
12:43square, or rectangular. Well, or
12:46Bartlett, exponential, Gauss, Hamming
12:49—careful, the Hamming window is
12:53different from the Hann window—
12:56Rosenfeld and so on, various Blackman
13:00windows. Well, a lot, isn't it? And
13:06then there are also the window
13:08parameters. It’s not enough to say,
13:12we have a Blackman window. Yes, we do
13:16too, but why do we get different
13:18results then? Because for Blackman
13:22windows, a parameter is specifically
13:25required here, or needs to be precisely
13:27defined. Which one should we use then?
13:32I really like using Kaiser-Bessel
13:34windows; they all have a parameter that
13:37must be set. There are often
13:39misunderstandings when manufacturers
13:42and customers agree on limit values,
13:45spectral limit values, and then later
13:48during acceptance or incoming
13:50inspection, it's said that your product
13:54does not meet the specifications. But
13:57we measured everything. Yes, how are
13:59you measuring? And then it turns out
14:01that different parameters were being
14:03used after all, and that’s why it’s
14:06important to know this if you want to
14:08perform accurate analyses. If you are a
14:12musician and just say, my god, it
14:13doesn't need to be that precise, I just
14:15want to know roughly what the spectrum
14:17looks like, then it's not that
14:19dangerous. Then I would recommend
14:22taking the Kaiser-Bessel window,
14:25varying the parameter a bit, seeing how
14:28the results differ, and if the
14:30differences are small, then that is
14:33okay. With good analyzers—and that
14:39one was good, I developed it at my
14:41measurement equipment company back then
14:43—you can choose different windows.
14:47Hann window, for example. You can
14:52select a parameter for the windows.
14:55Here now, Blackman-Harris window
14:58sidelobe attenuation. All windows lead
15:03to side lobes in the spectral range,
15:06and their attenuation is selected here.
15:13Then there are special settings like
15:16zero padding. This means you analyze a
15:22function, this time function, place a
15:26window over it, but then add zeros at
15:30the end, effectively increasing the
15:34window to that length. Naturally, no
15:38information is added in the time domain
15:41by doing this. And if no information is
15:44added in the time domain, then
15:46logically no information can be added
15:49in the frequency domain either. What is
15:52the effect? Zero padding 0, that's the
15:56case where nothing is added. A rather
16:00coarsely gridded spectrum. Here in the
16:04frequency domain, there is coarse
16:06gridding, coarse frequency
16:07discretization. Zero padding 2. Now
16:11it’s becoming a bit finer in the
16:13frequency domain. Zero padding 16. Now
16:16we have a really nice continuous
16:19function here, but it was only created
16:22through interpolation. You could have
16:26done that directly in the frequency
16:28domain with a good interpolation
16:30algorithm. No information is added.
16:33Knowing that is very important. This
16:37means such spectra will always contain
16:40artifacts—compromises that arise
16:43because you cannot optimize both time
16:46and frequency resolution simultaneously
16:50. The second is the overlap. This is a
16:57weighting window here that is placed
17:00successively at different positions
17:02over the time function to be analyzed.
17:07And you can say that when the first
17:09window ends, the next analysis starts,
17:12and we shift that by exactly one window
17:14width. But you can also say we shift it
17:18by less than a window width, so that
17:21there is an overlap here. As an example
17:25, this beat is switched on here and
17:29analyzed with such windows. Well, the
17:35first window doesn't catch anything yet
17:37, then this window comes, and
17:38unfortunately, these windows open right
17:40at the moment when the beat is at its
17:42minimum, which is unfavorable. If you
17:46work with overlap here, it already
17:49looks better. Here it is again with the
17:54result. That is the window, that is the
17:57beat, that is the spectral analysis
17:59with a Kaiser window. N is the number
18:02of points. In audio engineering, you
18:07like to work with 4096 here, but also
18:10sometimes 2*4 or divided by 2, 48. You
18:15have to see which analysis duration is
18:17suitable. It analyzes with 60%overlap,
18:23no zero padding. Time runs from front
18:30to back here. I would say that this
18:35beat is not clearly recognizable here.
18:41It is better if you don't just use
18:44these windows, but turn on even more
18:48overlap. 80%overlap, still not
18:54convincing. 93%overlap is getting
19:03better. The frequency resolution is
19:06still too poor. Zero padding 4. Now you
19:10can slowly start to see how this beat
19:13progresses over time and at what
19:16frequency. But there is still this
19:19leakage, this broadening in the
19:21frequency domain. Once again,
19:24simultaneous optimization of time and
19:28frequency resolution is not possible. A
19:32short time resolution with a sharp
19:35frequency resolution at the same time.
19:38That's not possible. In terms of system
19:41theory, that's not possible. It's not
19:43because the software is bad. Another
19:48example, a sinusoidal frequency
19:50modulation. We see a small section up
20:00here. The frequency of this sine tone
20:03increases and then decreases again. I
20:06can only display a small section of
20:08that. Important, the amplitude remains
20:12constant. And that is now being
20:15analyzed with such a spatial spectrum.
20:19Time from back to front, frequency from
20:22left to right, once with a Hann window
20:25and once with a Flat Top window. Both
20:32windows, both analyses are performed on
20:35the exact same time signal, and yet you
20:39can see clear differences here. What is
20:44especially astonishing is that peaks
20:47form at the turning points. At that
20:52point in time or in that time range
20:53where the frequency is highest, there
20:55is a maximum here. Then the amplitude
21:00apparently goes back down, goes back up
21:03, goes back down, then goes back up
21:05there again. And this can be seen here
21:07as well. However, the differences here
21:09are not as large. Yet, the amplitude in
21:13the time domain remains constant. This
21:17is not a software error. It has to be
21:20this way. As I’ve said before, I
21:25cannot go into every detail, Parseval's
21:28theorem, etc. Where the spectrum is
21:30high, it is very narrow, and where it
21:33goes down, the spectrum becomes wide.
21:36It must be that way. Now, let's switch
21:39over. Someone might come along and say
21:42this is nonsense. Why look up to 16 kHz
21:44if everything is happening down there?
21:48Just switch the frequency display. So,
21:53I have always specified the 2 kHz.
21:58Again, peaks here and here, two
22:01different windows: Hanning, Flat-top 80
22:05, meaning 80 dB side lobe attenuation.
22:10Side lobe attenuation. One more step
22:15further. And now it is slowly starting
22:20to get strange. Now these small jagged
22:23edges are appearing here. Also quite
22:27interesting. At the 200-millisecond
22:30mark, this spectrum in black looks like
22:35this, and this spectrum like that. One
22:40more step further, and now it becomes
22:45very clear; we are approaching a line
22:48spectrum. I will not go through the
22:50calculation. The Bessel spectrum, the
22:53FM spectrum, the frequency modulation
22:56spectrum—that is the long-term
22:58spectrum. It does not change anymore.
23:05At this point, one must know that if
23:08you narrow the spectral display—
23:11meaning you show a smaller frequency
23:14range—the time range over which you
23:17are analyzing automatically increases.
23:22And if you were to take this even
23:24further, you would eventually be
23:27analyzing over a very long time, and
23:30that is the long-term spectrum. There
23:36is only one long-term spectrum, but an
23:39infinite number of short-term spectra.
23:43These infinite numbers are defined by
23:47the analysis parameters: which sampling
23:50frequency, how many points, which
23:53weighting window, which FFT window.
23:58Possibly, it was set for zero-padding
24:02and with soft overlap. These parameters
24:06can be set in very different ways, and
24:09that is why there can be very different
24:12spectra, all of which are correct. You
24:16cannot simply say that one spectrum is
24:18incorrectly set. What is wrong and what
24:21is right always depends on what you
24:23want to see. And that is why one of the
24:26first questions is: what should this
24:28analysis yield? The parameter selection
24:32is based on that. Other representations
24:39of this frequency modulation: shown
24:41here in color, and here a view from
24:44above of this spectrum. You can still
24:48recognize that the frequency goes up
24:51here, then down again, then up again,
24:53then down again. That is frequency
24:57modulation. But you can also already
25:01see that lines are forming here, and as
25:04just mentioned, if you choose an even
25:07larger section in the time domain, the
25:10line structure becomes clearer and this
25:13back-and-forth in the time domain
25:15disappears more and more. And if you
25:20analyze over a very long period of time
25:23, then you get the long-term spectrum.
25:27What is all this for? I just said,
25:31before you set the parameters for a
25:34spectral analysis, you have to know
25:36what this analysis is for. From the
25:42spectral distribution of spectral
25:45maxima, so-called formants, one can get
25:48clues about the timbre. You need some
25:52experience for that. From the temporal
25:57progression of individual partial
26:00levels, you get clues about the sustain
26:03or decay. If the levels of harmonics
26:07drop off quickly. That can be a sign of
26:11old strings. The sound becomes dull.
26:15When hearing this adjective, you
26:17probably think first of a dull knife or
26:19scissors. But the term also exists in
26:23painting as dull colors, and certainly
26:26in acoustics as well. The sound is dull
26:29. The decay rate of the partial levels
26:33provides clues there. Is the spectrum
26:37harmonic or inharmonic? That is more
26:41about vibration engineering,
26:43mathematical descriptions of vibrations
26:45. In the spectrum, you can perhaps
26:49separate useful signals from
26:51interference. Interference can be noise
26:54or hum, but perhaps also another
26:56instrument. Then a spectral analysis
27:01can provide clues about the pitch, but
27:04here too you need experience, because
27:07the pitch of an instrument sound is
27:10generally not determined by the
27:12fundamental frequency. I will say a few
27:16words about that in psychoacoustics.
27:19Conclusion: At this point: The analysis
27:22of an instrument sound is relatively
27:24simple. "Relatively" here implies that
27:27you, of course, need experience with
27:30the chosen type of analysis, such as
27:32how to set the parameters. The
27:35interpretation, however, can become
27:38difficult. With good training, the
27:41measurement technology is not the
27:42problem. When it comes to interpreting
27:45the measurement data, that is where it
27:48can actually get difficult. A few
27:53examples, tone spectra, a low note, a
27:59high note. The first harmonic, second,
28:03third, fourth harmonic. Levels are
28:07shown here; when it comes to the
28:10relationship, i.e., the level
28:12differences, then it is not so
28:14important what 0 dB is. If you want it
28:19absolute, you can probably find that in
28:21this program as well. What bothers me
28:25here, and what I wouldn't do, is that
28:27he uses a rectangular window. In my
28:31view, the worst window, unless you are
28:35analyzing very specific signals. For
28:38impulse analyses, it can make sense.
28:42There are a few special cases, but for
28:44a general analysis of an instrument
28:47tone, I would never use a rectangular
28:49window. Up here at the peak, there can
28:53be level errors of up to 3.9 dB. Again,
28:57why that is, is not possible here; it
29:00would lead too far. It is the way it is
29:02. The leakage down here is already
29:05quite broad. Now one could say, well,
29:09if I'm only interested in the maxima up
29:11there and if I can accept a 3.9 dB
29:13level error, it wouldn't be so bad. But
29:16I wonder, what is all this junk down
29:19there or here for? Well, he writes that
29:23everything below 100 Hertz is unusable.
29:27That is a large part of the spectrum.
29:29Why represent it then? He doesn't even
29:32know exactly where it comes from,
29:33whether from the room or the
29:35electronics. It could well be that
29:38there is crosstalk in the computer from
29:41some RF oscillations that then mix down
29:44. No idea. Or the fan, but if you
29:49publish something like this on the
29:51internet, I wouldn't include something
29:54like that in the image. So either
30:00measure the spectrum only from higher
30:03frequencies and with the right window,
30:06or go into a quiet room, 20 m
30:08microphone cable and into the next room
30:11where no fan is running, or insert a
30:13high-pass filter and filter out the
30:16interference. But the best thing is to
30:21at least not go down to 3 Hertz here.
30:26That makes no sense in such an analysis
30:27. My fear is that this is an unusable
30:31program where you can set too little,
30:33which is also unfavorable. Here he has
30:3860 dB, here he has 45 dB, so it is hard
30:42to compare. It would be better if the
30:45dynamic range were always the same. And
30:49another curiosity, has everyone noticed
30:53this, if you look here or here, there
31:01is a line missing. That is not 10,000
31:05Hz, that is 9,000 Hz. The programmer
31:09saved himself the 10,000 Hz line. This
31:14problem is often found, that people who
31:17come from information processing,
31:20perhaps mathematicians, create such
31:23programs, but know relatively little
31:26about the application and then possibly
31:29make simplifications that are not
31:32always helpful. A few more spectra.
31:38First, second, third, fourth harmonic.
31:41It says A. I interpret that as
31:43amplitude linear representation. Okay,
31:47the fourth partial, the fourth harmonic
31:50is the largest here 1 2 3 4 not
31:53necessarily here. Which doesn't mean
31:57this spectrum has to be wrong. It also
32:00doesn't mean this spectrum has to be
32:01wrong. It means such instrumental
32:05sounds cannot be represented with a
32:09single spectrum. The spectra depend on
32:14which note is being played. They even
32:17depend on where the microphone is. And
32:20then these spectra are time-variant.
32:24Especially with impulsive sounds like
32:26the electric guitar produces, these
32:28levels change over time. You can derive
32:34certain criteria here, for example that
32:37the even-numbered harmonics, the second
32:40, fourth, etc., are also smaller in
32:42their amplitude there than the
32:45odd-numbered ones. This results from
32:49the resonances in this blowpipe. But
32:53one has to be very careful when taking
32:55such a picture and saying: "Aha, that
32:57is the overtone spectrum" or that one
32:59too. It's not that simple. A different
33:07image, a different scale. The question
33:11starts right here: where do the 10 dB
33:14belong? I'd guess a bit off-center,
33:18further up. So those would be 10 dB,
33:22that's 0 dB, then that's -80 and here
33:25-90 dB. An impressive dynamic range of
33:30100 dB, with everything down here being
33:33junk. It would be better to limit the
33:37dynamic range to a smaller area. That's
33:41what I would have done, at least. 22 Hz
33:44, presumably here or in the middle of
33:47this label; there are also analyzers
33:50where you have to take the middle of
33:53the frequency notation. I suspect
33:57that's where 22 is. Now I've rounded
34:02the factor a bit to show how you get
34:04from one frequency to the next. That's
34:08roughly a factor of 2, 2, 2. Now
34:13suddenly it goes from just under 700 Hz
34:15to 1 kHz. Now the label has a factor of
34:201.45 there. From one it goes to three
34:25and then it's roughly a factor of 2
34:27again. Why it doesn't say 12 kHz there,
34:31I don't know. And the reason is most
34:35likely that it's rounding here, and now
34:39it says 1 kHz. But it's probably double
34:45the value, so roughly 1.4. You don't
34:49know for sure that the 'k' should
34:52actually be lowercase, but let's let
34:55that slide, you'd understand it, but I
34:58don't understand this scaling in that
35:01range. Once again, these aren't
35:05frequencies down here, but
35:06relationships. The next question: What
35:10about this frequency labeling? There
35:13are standard frequencies in acoustics.
35:17These are not standard frequencies. Did
35:20some programmer just think something up
35:23? Not so good. Another example, a piano
35:29spectrum, which was well-intentioned.
35:34For the partial tone levels, it has
35:36given the numerical values here to a
35:38thousandth of a dB. That's nonsense. If
35:43you're familiar with acoustics, you
35:46know that I have to round it myself now
35:48. 82.046 dBA. No analyzer, no
35:54microphone can measure to a thousandth
35:57of a dB. So it shouldn't display it. A
36:02tenth of a dB is perfectly sufficient
36:05here because neither the microphone nor
36:08the components used in the analog
36:10section have significantly higher
36:13precision. Then the question arises
36:18again: why display all this junk here
36:21with a dynamic range of a whopping 130
36:24dB if you ultimately only want to show
36:27what's up there? I can certainly glean
36:31information from it, but the danger is
36:34that someone who doesn't have much
36:36experience yet will start trying to
36:38interpret all of it as well. I would
36:42have left that out. I would have taken
36:44a section of what's up there. One more
36:47supplementary explanation. This is a
36:50one-twelfth octave analysis. This is
36:53not an FFT. I will come back to octave
36:59and 1/3 octave analyses later when we
37:04are down there. Where exactly is 10,000
37:08Hz, where is 1,000? You can find the
37:16assignment if you look long enough. In
37:19a previous image, the assignment was on
37:20the far left; now it's in the middle of
37:22that number. Well. You have to know
37:25that. And another thing you have to
37:29know: with such an analysis, the
37:31absolute bandwidths increase as you go
37:33higher. Absolute bandwidth in Hertz,
37:37the difference between the upper and
37:39lower cutoff frequency. Down here, the
37:43absolute bandwidth is only about one
37:45Hertz. That means these frequency bands
37:48take a correspondingly long time to
37:51settle, the reciprocal is one second,
37:53and if you want to measure precisely,
37:56you take three or four times that. So,
38:00what you also need to know about such
38:021/3 octave analysis is that up here you
38:05have very fast settling times, down
38:07here you have slow ones, and then the
38:09questions arise: okay, we have to
38:11average, how is it done? Linear,
38:14exponential, and so on. So, expert
38:17knowledge is required there too.
38:20Another image; the graph shows the
38:23level and frequency ratios. Frequency,
38:26okay, take the center, standard
38:28frequencies, good, but here level in dB
38:34, there’s nothing labeled. For a very
38:39rough orientation, that might be
38:41sufficient. But you can't gather much
38:45from an image like that. Could be done
38:47better. Finally, Mr. Clapton from the
38:52track "Stepping Out." An early
38:55recording with John Mayall. For these
38:58recordings, the G3 note was analyzed,
39:01which appears several times in the
39:03piece. Here at four different points.
39:08It’s always the same pitch at
39:09different times. This is the middle of
39:12the evaluation window, and you can see
39:15that even though it’s always the same
39:18Mr. Clapton playing, with the same
39:20guitar and most likely the same
39:22settings on the guitar and amplifier.
39:26It looks different. You can’t just
39:29say, "That is the Clapton spectrum."
39:32Even if you refer to this G3 note, well
39:36, he also knows that through small
39:39variations with the plectrum, the angle
39:42at which he strikes the strings, the
39:45position where he hits them, and how he
39:49strikes them, he can influence the
39:52sound. You can guess roughly that
39:56there’s a minimum, a minimum between
39:591 and 2 kHz. You can see that here, and
40:03here, and here. But the exact curve,
40:07the relation of the individual levels,
40:10that is different. Therefore, spectra
40:14can help with sound analysis. They are
40:19also easy to create if you know how.
40:22But the interpretation can then become
40:25difficult. So, I need to do the
40:28third-octave analysis quickly so it
40:30doesn't take too long. With standard
40:33Fourier analysis, we get a linear
40:36frequency axis; with standard
40:38third-octave analysis, a logarithmic
40:41one. You can change these analysis
40:44results later on, but for now, we'll
40:46stick to the standard
40:48linear-logarithmic display. For our
40:52hearing, the logarithmic frequency axis
40:54is often better, because with the 100
40:57Hz, or 0.1 kHz, down there in the
41:00linear view, you can't really resolve
41:03the low-frequency range. Analysis is
41:08performed using third-octave filters, a
41:10filter bank where the ratio of the
41:12upper cutoff frequency to the lower
41:14cutoff frequency is a major third. For
41:17musicians, a major third is four
41:19semitones. And now, these filters are
41:23used to see how much power falls into
41:25each individual filter. There are
41:29standards, various standards with
41:31different accuracy requirements.
41:33Standards in Europe, and in America,
41:35there are other standards again. Anyone
41:38who needs this professionally must
41:40first work their way through the
41:42regulations. This is what third-octave
41:44analyses can look like. The frequency,
41:47the time from back to front. Here is
41:50the response of a room. At zero, it's
41:54excited with a bang. The high-frequency
41:56levels drop off quickly, the
41:58low-frequency ones drop off slowly. The
42:00reverberation time is then calculated
42:02from this decay, for example. A color
42:05spectrogram. You can see that it is in
42:09the frequency range of Hertz. Here it
42:12goes upwards on a logarithmic frequency
42:15axis. It is roughly gridded in the
42:17frequency domain. At a specific point
42:20in time, for example here, you can look
42:22at a third-octave spectrum. Clearly,
42:25this large filter bandwidth is
42:28relatively wide in this case. That is a
42:32third-octave width, or one-third of an
42:34octave. If you want half the filter
42:38bandwidth, you use sixth-octave,
42:40twelfth-octave, or twenty-fourth-octave
42:43filters; then the resolution becomes
42:46finer, but the time resolution becomes
42:49worse accordingly. You can realize such
42:54bandpass filters relatively easily with
42:56digital filters. I did that with three
42:59Butterworth filters. Today, you might
43:01use four or five such filters. Here it
43:05is drawn for three Butterworth filters,
43:08and there is the problem: the closer
43:11the filter center frequency gets to the
43:14Nyquist frequency—in this case 24 kHz
43:17—the more a tangent function makes
43:20itself felt. Nothing more precise at
43:23this point, just a note. This means you
43:26deviate from the target frequency
43:28response; the filters get steeper up
43:31here, which is not necessarily an
43:33advantage if the rule is that filters
43:36must stay within a certain band. Well,
43:40in that case, it wouldn't be a big
43:41problem; it can be as steep as it wants
43:43as long as it doesn't cross the limit
43:45line here. However, there are other
43:47standards and other regulations. You
43:49have to look those up exactly.
43:51Third-octave analysis used to be the
43:53domain of analog technology. In boxes
43:56like these, analog filters were
43:58installed, which is why the thing was
43:59so big. There were also small ones to
44:03attach to a handheld sound level meter.
44:07It was also available digitally at
44:10steep prices 65581 D-Marks or here
44:14nearly 100,000. Well, by now you'd
44:18almost have to convert that one-to-one.
44:22If you bought an FFT and a third-octave
44:25analyzer back then, you'd be near €
44:28200,000 in today's money. One advantage
44:34of these devices, assuming they aren't
44:37broken by now, you switch them on and
44:40get an analysis. You don't have to wait
44:43for the computer to boot. You don't
44:45have to find out there's another update
44:47waiting. All the settings are gone. You
44:51don't have to worry that after an
44:53update, your old data can no longer be
44:55read. I still use this filter to this
44:59day. You switch it on and you can
45:02filter. Can be an advantage. I also use
45:07digital devices. If you want to save or
45:10print the results, it is indeed an
45:12advantage. when you can use digital
45:16signal processing. Well, actually, a
45:20section on psychoacoustics was planned
45:23for this point, but it would just get
45:26too long again. That's why I'm moving
45:30the psychoacoustics part that belongs
45:32here to the next video. A very brief
45:37summary at this point. The first part
45:42showed that perceptions are subjective
45:45and can be influenced. The second part
45:50showed that measurements are objective
45:54but need the right parameters for the
45:58issue at hand and require
46:01interpretation, which can sometimes be
46:04difficult. More on that in the next
46:08video. Subscribe to the channel so you
46:13don't miss any. is.

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