On 09/22/2012 01:52 PM, Fons Adriaensen wrote: > On Sat, Sep 22, 2012 at 04:19:48AM +0200, Robin Gareus wrote: > >> Fons - author of JAAA and JAPA - is on this list and may chime in sooner >> or later. > > Eccomi. Ecce Fonso. Wow. Many Thanks for this explanation: > Confusion as to what JAPA actually measures is a recurring thing... > Unfortunately it's not that easy to explain without going into a > bit of theory. > > Any spectrum analyser is in the end just a set of bandpass filters > acting on the input signal. The outputs levels of these filters are > then displayed as a function of frequency. > > The differences are about how these filters are distributed over > the audio range. In all cases we'll assume that together they cover > this range, and that they overlap in a 'sensible' way, for example > the filter curves intersect at the -3dB points [1]. So the distances > between the center frequencies and the bandwidth of the filters > are related. > > A second thing to consider is the nature of the signal that is being > measured. This could have a line spectrum, i.e. consist of a set of > discrete frequencies (sine waves), or it could be a noise-like signal, > or a mix of the two. The point about noise signals is that their energy > is not concentrated into single frequencies but distributed over a > continuous frequency range. If you could measure them at exactly one > single frequency (not possible, it would take infinite time), you > would find zero. But if you measure them over a finite frequency > interval you find a non-zero value. Noise is characterized by its > _density_, that is the power per Hz. > > White noise has the same density at all frequency (within some range). > There is as much energy between say 5000 and 5010 Hz as there is between > 100 and 110 Hz, or 20 and 30 Hz etc. If you send white noise through two > bandfilters, one with a bandwidth B and one with a bandwidth 2 * B, then > the output level of the second one would be 3 dB (a factor of 2 in power) > higher than the first one. > > Pink noise has a density that is inversely proportional to frequency. > That means that if you integrate over an interval corresponding to > some fixed _ratio_ (rather than difference) of frequencies, you find > the same value. For example there is as much power between 1000 and > 2000 Hz as there is between 100 and 200 Hz, or between 10 and 20 Hz. > > Returning to the analyser, the simplest case is a set of filters that > all have the same bandwidth (measured in Hz, not octaves) and the same > gain at their center frequencies. That's the case for e.g. JAAA. Since > all filters have the same gain, sine waves will be measured correctly. > White noise will produce the same output level for all filters, and > be displayed as a flat trace. Pink noise will result in a trace that > goes down by 3 dB per octave, since its density is proportional to the > inverse of frequency. > > Now imagine the set of filters used in e.g. a 1/3 octave analyser. > Center frequencies are a factor of around 1.26 apart, e.g. 100, 125, > 160, 200, 250, 315, etc [2]. The bandwidths of the filters increase in > the same way - they are proportional to the center frequencies instead > of being all the same. The filter centered at 1 kHz is ten times as wide > (in Hz) as the one at 100 Hz. If all filters have again the same gain, > then sine waves (at the center frequencies) will be measured correctly. > But now, since the bandwidths increase with frequency, white noise > will appear as spectrum that rises +3dB / octave, and pink noise will > be shown as a flat spectrum. > > What this shows is that, at least for noise-like signals, or when > you are not interested in single frequencies but more in the general > shape of the spectrum, there is no single 'correct' way to show it, > it's a matter of interpretation. Which one of the two above is the > more relevant depends on the application [3]. > > The filter set used by JAPA is something in between the two shown > above. At least in the medium frequency range, the filter bandwidths > are proportional to the 'critical bandwidths' of the human hearing > mechanism [4]. You can get an idea of how the filters are distributed > by selecting the 'warped' frequency scale. With this option, all > filter have the same width *on the display*. You will see that the > very low and very high frequency ranges are 'compressed' compared > to a logarithmic scale, there are less filters there, while the > resolution in the mid frequency range is increased. How exactly > this is done depends on the 'warp factor' which you can select > on the right panel. The same filters are used if you select the > normal logarithmic scale, only the display is different. > > With the response set to 'flat', all filters have the same gain. > So a slow sine sweep would produce a flat trace. But since the > filter bandwidts are neither constant nor proportional to frequency, > neither white nor pink noise will be shown as a flat spectrum. > > What happens if you select the 'prop' response is that the filter > _gains_ are modified so you get a flat trace for pink noise. The > consequence is that sine waves will not be measured correctly, so > it depends on your application which response makes sense. > > Ciao, > > [1] In both JAAA and JAPA there are actually about twice as much > filters as would be suggested by this, but that doesn't change > the principle. > > [2] Actually 1/3 octave is a misnomer, all real-life analysers > use a ratio of 1/10 decade in order to have a set of 'round' > center frequencies including e.g. 100, 1k, 10k. > 10^(1/10) = 1.2589... while 2^(1/3) = 1.2599... > > [3] This also means that if you would modify e.g. JAPA to have > a log frequency scale it would still be the same analyser, it > will still show -3 dB/octave for pink noise. Which is not what > one would expect from a 'log' analyser. > > [4] This is not directly related to Fletcher-Munson or other > equal loudness curves. The critical bandwidths are about > masking, that is to what extent one frequency can hide the > presence of another, which in turn is related to what level > of detail in a spectrum is relevant to human hearing. > _______________________________________________ Linux-audio-user mailing list Linux-audio-user@xxxxxxxxxxxxxxxxxxxx http://lists.linuxaudio.org/listinfo/linux-audio-user
